Exact Binomial Coefficients, Complete Expansions and Natural-Exponent Boundaries

(0, 0, 1, 1, Yes)

Compute exact C(n,k) and C(n,n−k) and their equality. These count choices of k y-factors from n factors, including both endpoints and n=0. The exact result, retaining the requested order, is (0, 0, 1, 1, Yes).

(1, 1, 1, 1, Yes)

Compute exact C(n,k) and C(n,n−k) and their equality. These count choices of k y-factors from n factors, including both endpoints and n=0. The exact result, retaining the requested order, is (1, 1, 1, 1, Yes).

(2, 2, 1, 1, Yes)

Compute exact C(n,k) and C(n,n−k) and their equality. These count choices of k y-factors from n factors, including both endpoints and n=0. The exact result, retaining the requested order, is (2, 2, 1, 1, Yes).

(3, 3, 1, 1, Yes)

Compute exact C(n,k) and C(n,n−k) and their equality. These count choices of k y-factors from n factors, including both endpoints and n=0. The exact result, retaining the requested order, is (3, 3, 1, 1, Yes).

(4, 4, 1, 1, Yes)

Compute exact C(n,k) and C(n,n−k) and their equality. These count choices of k y-factors from n factors, including both endpoints and n=0. The exact result, retaining the requested order, is (4, 4, 1, 1, Yes).

(5, 5, 1, 1, Yes)

Compute exact C(n,k) and C(n,n−k) and their equality. These count choices of k y-factors from n factors, including both endpoints and n=0. The exact result, retaining the requested order, is (5, 5, 1, 1, Yes).

(6, 6, 1, 1, Yes)

Compute exact C(n,k) and C(n,n−k) and their equality. These count choices of k y-factors from n factors, including both endpoints and n=0. The exact result, retaining the requested order, is (6, 6, 1, 1, Yes).

(7, 7, 1, 1, Yes)

Compute exact C(n,k) and C(n,n−k) and their equality. These count choices of k y-factors from n factors, including both endpoints and n=0. The exact result, retaining the requested order, is (7, 7, 1, 1, Yes).

(8, 8, 1, 1, Yes)

Compute exact C(n,k) and C(n,n−k) and their equality. These count choices of k y-factors from n factors, including both endpoints and n=0. The exact result, retaining the requested order, is (8, 8, 1, 1, Yes).

(9, 9, 1, 1, Yes)

Compute exact C(n,k) and C(n,n−k) and their equality. These count choices of k y-factors from n factors, including both endpoints and n=0. The exact result, retaining the requested order, is (9, 9, 1, 1, Yes).

(10, 10, 1, 1, Yes)

Compute exact C(n,k) and C(n,n−k) and their equality. These count choices of k y-factors from n factors, including both endpoints and n=0. The exact result, retaining the requested order, is (10, 10, 1, 1, Yes).

(11, 11, 1, 1, Yes)

Compute exact C(n,k) and C(n,n−k) and their equality. These count choices of k y-factors from n factors, including both endpoints and n=0. The exact result, retaining the requested order, is (11, 11, 1, 1, Yes).

(12, 12, 1, 1, Yes)

Compute exact C(n,k) and C(n,n−k) and their equality. These count choices of k y-factors from n factors, including both endpoints and n=0. The exact result, retaining the requested order, is (12, 12, 1, 1, Yes).

(13, 13, 1, 1, Yes)

Compute exact C(n,k) and C(n,n−k) and their equality. These count choices of k y-factors from n factors, including both endpoints and n=0. The exact result, retaining the requested order, is (13, 13, 1, 1, Yes).

(14, 14, 1, 1, Yes)

Compute exact C(n,k) and C(n,n−k) and their equality. These count choices of k y-factors from n factors, including both endpoints and n=0. The exact result, retaining the requested order, is (14, 14, 1, 1, Yes).

(15, 15, 1, 1, Yes)

Compute exact C(n,k) and C(n,n−k) and their equality. These count choices of k y-factors from n factors, including both endpoints and n=0. The exact result, retaining the requested order, is (15, 15, 1, 1, Yes).

(16, 16, 1, 1, Yes)

Compute exact C(n,k) and C(n,n−k) and their equality. These count choices of k y-factors from n factors, including both endpoints and n=0. The exact result, retaining the requested order, is (16, 16, 1, 1, Yes).

(17, 17, 1, 1, Yes)

Compute exact C(n,k) and C(n,n−k) and their equality. These count choices of k y-factors from n factors, including both endpoints and n=0. The exact result, retaining the requested order, is (17, 17, 1, 1, Yes).

(18, 18, 1, 1, Yes)

Compute exact C(n,k) and C(n,n−k) and their equality. These count choices of k y-factors from n factors, including both endpoints and n=0. The exact result, retaining the requested order, is (18, 18, 1, 1, Yes).

(19, 19, 1, 1, Yes)

Compute exact C(n,k) and C(n,n−k) and their equality. These count choices of k y-factors from n factors, including both endpoints and n=0. The exact result, retaining the requested order, is (19, 19, 1, 1, Yes).

(20, 20, 1, 1, Yes)

Compute exact C(n,k) and C(n,n−k) and their equality. These count choices of k y-factors from n factors, including both endpoints and n=0. The exact result, retaining the requested order, is (20, 20, 1, 1, Yes).

(21, 21, 1, 1, Yes)

Compute exact C(n,k) and C(n,n−k) and their equality. These count choices of k y-factors from n factors, including both endpoints and n=0. The exact result, retaining the requested order, is (21, 21, 1, 1, Yes).

(22, 22, 1, 1, Yes)

Compute exact C(n,k) and C(n,n−k) and their equality. These count choices of k y-factors from n factors, including both endpoints and n=0. The exact result, retaining the requested order, is (22, 22, 1, 1, Yes).

(23, 23, 1, 1, Yes)

Compute exact C(n,k) and C(n,n−k) and their equality. These count choices of k y-factors from n factors, including both endpoints and n=0. The exact result, retaining the requested order, is (23, 23, 1, 1, Yes).

(24, 24, 1, 1, Yes)

Compute exact C(n,k) and C(n,n−k) and their equality. These count choices of k y-factors from n factors, including both endpoints and n=0. The exact result, retaining the requested order, is (24, 24, 1, 1, Yes).

(0, [[0, 1, 0, 0]], constant_one_boundary)

Every row is [k,C(n,k),x exponent n−k,y exponent k]. Retain all n+1 rows and exact integer coefficients. At n=0 the expansion is constant1, including the formal zero-exponent convention. Noncommuting variables require a different theorem. The exact result, retaining the requested order, is (0, [[0, 1, 0, 0]], constant_one_boundary).

(1, [[0, 1, 1, 0], [1, 1, 0, 1]], complete_k_zero_through_n_coefficients)

Every row is [k,C(n,k),x exponent n−k,y exponent k]. Retain all n+1 rows and exact integer coefficients. At n=0 the expansion is constant1, including the formal zero-exponent convention. Noncommuting variables require a different theorem. The exact result, retaining the requested order, is (1, [[0, 1, 1, 0], [1, 1, 0, 1]], complete_k_zero_through_n_coefficients).

(2, [[0, 1, 2, 0], [1, 2, 1, 1], [2, 1, 0, 2]], complete_k_zero_through_n_coefficients)

Every row is [k,C(n,k),x exponent n−k,y exponent k]. Retain all n+1 rows and exact integer coefficients. At n=0 the expansion is constant1, including the formal zero-exponent convention. Noncommuting variables require a different theorem. The exact result, retaining the requested order, is (2, [[0, 1, 2, 0], [1, 2, 1, 1], [2, 1, 0, 2]], complete_k_zero_through_n_coefficients).

(3, [[0, 1, 3, 0], [1, 3, 2, 1], [2, 3, 1, 2], [3, 1, 0, 3]], complete_k_zero_through_n_coefficients)

Every row is [k,C(n,k),x exponent n−k,y exponent k]. Retain all n+1 rows and exact integer coefficients. At n=0 the expansion is constant1, including the formal zero-exponent convention. Noncommuting variables require a different theorem. The exact result, retaining the requested order, is (3, [[0, 1, 3, 0], [1, 3, 2, 1], [2, 3, 1, 2], [3, 1, 0, 3]], complete_k_zero_through_n_coefficients).

(4, [[0, 1, 4, 0], [1, 4, 3, 1], [2, 6, 2, 2], [3, 4, 1, 3], [4, 1, 0, 4]], complete_k_zero_through_n_coefficients)

Every row is [k,C(n,k),x exponent n−k,y exponent k]. Retain all n+1 rows and exact integer coefficients. At n=0 the expansion is constant1, including the formal zero-exponent convention. Noncommuting variables require a different theorem. The exact result, retaining the requested order, is (4, [[0, 1, 4, 0], [1, 4, 3, 1], [2, 6, 2, 2], [3, 4, 1, 3], [4, 1, 0, 4]], complete_k_zero_through_n_coefficients).

(5, [[0, 1, 5, 0], [1, 5, 4, 1], [2, 10, 3, 2], [3, 10, 2, 3], [4, 5, 1, 4], [5, 1, 0, 5]], complete_k_zero_through_n_coefficients)

Every row is [k,C(n,k),x exponent n−k,y exponent k]. Retain all n+1 rows and exact integer coefficients. At n=0 the expansion is constant1, including the formal zero-exponent convention. Noncommuting variables require a different theorem. The exact result, retaining the requested order, is (5, [[0, 1, 5, 0], [1, 5, 4, 1], [2, 10, 3, 2], [3, 10, 2, 3], [4, 5, 1, 4], [5, 1, 0, 5]], complete_k_zero_through_n_coefficients).

(6, [[0, 1, 6, 0], [1, 6, 5, 1], [2, 15, 4, 2], [3, 20, 3, 3], [4, 15, 2, 4], [5, 6, 1, 5], [6, 1, 0, 6]], complete_k_zero_through_n_coefficients)

Every row is [k,C(n,k),x exponent n−k,y exponent k]. Retain all n+1 rows and exact integer coefficients. At n=0 the expansion is constant1, including the formal zero-exponent convention. Noncommuting variables require a different theorem. The exact result, retaining the requested order, is (6, [[0, 1, 6, 0], [1, 6, 5, 1], [2, 15, 4, 2], [3, 20, 3, 3], [4, 15, 2, 4], [5, 6, 1, 5], [6, 1, 0, 6]], complete_k_zero_through_n_coefficients).

(7, [[0, 1, 7, 0], [1, 7, 6, 1], [2, 21, 5, 2], [3, 35, 4, 3], [4, 35, 3, 4], [5, 21, 2, 5], [6, 7, 1, 6], [7, 1, 0, 7]], complete_k_zero_through_n_coefficients)

Every row is [k,C(n,k),x exponent n−k,y exponent k]. Retain all n+1 rows and exact integer coefficients. At n=0 the expansion is constant1, including the formal zero-exponent convention. Noncommuting variables require a different theorem. The exact result, retaining the requested order, is (7, [[0, 1, 7, 0], [1, 7, 6, 1], [2, 21, 5, 2], [3, 35, 4, 3], [4, 35, 3, 4], [5, 21, 2, 5], [6, 7, 1, 6], [7, 1, 0, 7]], complete_k_zero_through_n_coefficients).

(8, [[0, 1, 8, 0], [1, 8, 7, 1], [2, 28, 6, 2], [3, 56, 5, 3], [4, 70, 4, 4], [5, 56, 3, 5], [6, 28, 2, 6], [7, 8, 1, 7], [8, 1, 0, 8]], complete_k_zero_through_n_coefficients)

Every row is [k,C(n,k),x exponent n−k,y exponent k]. Retain all n+1 rows and exact integer coefficients. At n=0 the expansion is constant1, including the formal zero-exponent convention. Noncommuting variables require a different theorem. The exact result, retaining the requested order, is (8, [[0, 1, 8, 0], [1, 8, 7, 1], [2, 28, 6, 2], [3, 56, 5, 3], [4, 70, 4, 4], [5, 56, 3, 5], [6, 28, 2, 6], [7, 8, 1, 7], [8, 1, 0, 8]], complete_k_zero_through_n_coefficients).

(9, [[0, 1, 9, 0], [1, 9, 8, 1], [2, 36, 7, 2], [3, 84, 6, 3], [4, 126, 5, 4], [5, 126, 4, 5], [6, 84, 3, 6], [7, 36, 2, 7], [8, 9, 1, 8], [9, 1, 0, 9]], complete_k_zero_through_n_coefficients)

Every row is [k,C(n,k),x exponent n−k,y exponent k]. Retain all n+1 rows and exact integer coefficients. At n=0 the expansion is constant1, including the formal zero-exponent convention. Noncommuting variables require a different theorem. The exact result, retaining the requested order, is (9, [[0, 1, 9, 0], [1, 9, 8, 1], [2, 36, 7, 2], [3, 84, 6, 3], [4, 126, 5, 4], [5, 126, 4, 5], [6, 84, 3, 6], [7, 36, 2, 7], [8, 9, 1, 8], [9, 1, 0, 9]], complete_k_zero_through_n_coefficients).

(10, [[0, 1, 10, 0], [1, 10, 9, 1], [2, 45, 8, 2], [3, 120, 7, 3], [4, 210, 6, 4], [5, 252, 5, 5], [6, 210, 4, 6], [7, 120, 3, 7], [8, 45, 2, 8], [9, 10, 1, 9], [10, 1, 0, 10]], complete_k_zero_through_n_coefficients)

Every row is [k,C(n,k),x exponent n−k,y exponent k]. Retain all n+1 rows and exact integer coefficients. At n=0 the expansion is constant1, including the formal zero-exponent convention. Noncommuting variables require a different theorem. The exact result, retaining the requested order, is (10, [[0, 1, 10, 0], [1, 10, 9, 1], [2, 45, 8, 2], [3, 120, 7, 3], [4, 210, 6, 4], [5, 252, 5, 5], [6, 210, 4, 6], [7, 120, 3, 7], [8, 45, 2, 8], [9, 10, 1, 9], [10, 1, 0, 10]], complete_k_zero_through_n_coefficients).

(11, [[0, 1, 11, 0], [1, 11, 10, 1], [2, 55, 9, 2], [3, 165, 8, 3], [4, 330, 7, 4], [5, 462, 6, 5], [6, 462, 5, 6], [7, 330, 4, 7], [8, 165, 3, 8], [9, 55, 2, 9], [10, 11, 1, 10], [11, 1, 0, 11]], complete_k_zero_through_n_coefficients)

Every row is [k,C(n,k),x exponent n−k,y exponent k]. Retain all n+1 rows and exact integer coefficients. At n=0 the expansion is constant1, including the formal zero-exponent convention. Noncommuting variables require a different theorem. The exact result, retaining the requested order, is (11, [[0, 1, 11, 0], [1, 11, 10, 1], [2, 55, 9, 2], [3, 165, 8, 3], [4, 330, 7, 4], [5, 462, 6, 5], [6, 462, 5, 6], [7, 330, 4, 7], [8, 165, 3, 8], [9, 55, 2, 9], [10, 11, 1, 10], [11, 1, 0, 11]], complete_k_zero_through_n_coefficients).

(12, [[0, 1, 12, 0], [1, 12, 11, 1], [2, 66, 10, 2], [3, 220, 9, 3], [4, 495, 8, 4], [5, 792, 7, 5], [6, 924, 6, 6], [7, 792, 5, 7], [8, 495, 4, 8], [9, 220, 3, 9], [10, 66, 2, 10], [11, 12, 1, 11], [12, 1, 0, 12]], complete_k_zero_through_n_coefficients)

Every row is [k,C(n,k),x exponent n−k,y exponent k]. Retain all n+1 rows and exact integer coefficients. At n=0 the expansion is constant1, including the formal zero-exponent convention. Noncommuting variables require a different theorem. The exact result, retaining the requested order, is (12, [[0, 1, 12, 0], [1, 12, 11, 1], [2, 66, 10, 2], [3, 220, 9, 3], [4, 495, 8, 4], [5, 792, 7, 5], [6, 924, 6, 6], [7, 792, 5, 7], [8, 495, 4, 8], [9, 220, 3, 9], [10, 66, 2, 10], [11, 12, 1, 11], [12, 1, 0, 12]], complete_k_zero_through_n_coefficients).

(13, [[0, 1, 13, 0], [1, 13, 12, 1], [2, 78, 11, 2], [3, 286, 10, 3], [4, 715, 9, 4], [5, 1287, 8, 5], [6, 1716, 7, 6], [7, 1716, 6, 7], [8, 1287, 5, 8], [9, 715, 4, 9], [10, 286, 3, 10], [11, 78, 2, 11], [12, 13, 1, 12], [13, 1, 0, 13]], complete_k_zero_through_n_coefficients)

Every row is [k,C(n,k),x exponent n−k,y exponent k]. Retain all n+1 rows and exact integer coefficients. At n=0 the expansion is constant1, including the formal zero-exponent convention. Noncommuting variables require a different theorem. The exact result, retaining the requested order, is (13, [[0, 1, 13, 0], [1, 13, 12, 1], [2, 78, 11, 2], [3, 286, 10, 3], [4, 715, 9, 4], [5, 1287, 8, 5], [6, 1716, 7, 6], [7, 1716, 6, 7], [8, 1287, 5, 8], [9, 715, 4, 9], [10, 286, 3, 10], [11, 78, 2, 11], [12, 13, 1, 12], [13, 1, 0, 13]], complete_k_zero_through_n_coefficients).

(14, [[0, 1, 14, 0], [1, 14, 13, 1], [2, 91, 12, 2], [3, 364, 11, 3], [4, 1001, 10, 4], [5, 2002, 9, 5], [6, 3003, 8, 6], [7, 3432, 7, 7], [8, 3003, 6, 8], [9, 2002, 5, 9], [10, 1001, 4, 10], [11, 364, 3, 11], [12, 91, 2, 12], [13, 14, 1, 13], [14, 1, 0, 14]], complete_k_zero_through_n_coefficients)

Every row is [k,C(n,k),x exponent n−k,y exponent k]. Retain all n+1 rows and exact integer coefficients. At n=0 the expansion is constant1, including the formal zero-exponent convention. Noncommuting variables require a different theorem. The exact result, retaining the requested order, is (14, [[0, 1, 14, 0], [1, 14, 13, 1], [2, 91, 12, 2], [3, 364, 11, 3], [4, 1001, 10, 4], [5, 2002, 9, 5], [6, 3003, 8, 6], [7, 3432, 7, 7], [8, 3003, 6, 8], [9, 2002, 5, 9], [10, 1001, 4, 10], [11, 364, 3, 11], [12, 91, 2, 12], [13, 14, 1, 13], [14, 1, 0, 14]], complete_k_zero_through_n_coefficients).

(15, [[0, 1, 15, 0], [1, 15, 14, 1], [2, 105, 13, 2], [3, 455, 12, 3], [4, 1365, 11, 4], [5, 3003, 10, 5], [6, 5005, 9, 6], [7, 6435, 8, 7], [8, 6435, 7, 8], [9, 5005, 6, 9], [10, 3003, 5, 10], [11, 1365, 4, 11], [12, 455, 3, 12], [13, 105, 2, 13], [14, 15, 1, 14], [15, 1, 0, 15]], complete_k_zero_through_n_coefficients)

Every row is [k,C(n,k),x exponent n−k,y exponent k]. Retain all n+1 rows and exact integer coefficients. At n=0 the expansion is constant1, including the formal zero-exponent convention. Noncommuting variables require a different theorem. The exact result, retaining the requested order, is (15, [[0, 1, 15, 0], [1, 15, 14, 1], [2, 105, 13, 2], [3, 455, 12, 3], [4, 1365, 11, 4], [5, 3003, 10, 5], [6, 5005, 9, 6], [7, 6435, 8, 7], [8, 6435, 7, 8], [9, 5005, 6, 9], [10, 3003, 5, 10], [11, 1365, 4, 11], [12, 455, 3, 12], [13, 105, 2, 13], [14, 15, 1, 14], [15, 1, 0, 15]], complete_k_zero_through_n_coefficients).

(16, [[0, 1, 16, 0], [1, 16, 15, 1], [2, 120, 14, 2], [3, 560, 13, 3], [4, 1820, 12, 4], [5, 4368, 11, 5], [6, 8008, 10, 6], [7, 11440, 9, 7], [8, 12870, 8, 8], [9, 11440, 7, 9], [10, 8008, 6, 10], [11, 4368, 5, 11], [12, 1820, 4, 12], [13, 560, 3, 13], [14, 120, 2, 14], [15, 16, 1, 15], [16, 1, 0, 16]], complete_k_zero_through_n_coefficients)

Every row is [k,C(n,k),x exponent n−k,y exponent k]. Retain all n+1 rows and exact integer coefficients. At n=0 the expansion is constant1, including the formal zero-exponent convention. Noncommuting variables require a different theorem. The exact result, retaining the requested order, is (16, [[0, 1, 16, 0], [1, 16, 15, 1], [2, 120, 14, 2], [3, 560, 13, 3], [4, 1820, 12, 4], [5, 4368, 11, 5], [6, 8008, 10, 6], [7, 11440, 9, 7], [8, 12870, 8, 8], [9, 11440, 7, 9], [10, 8008, 6, 10], [11, 4368, 5, 11], [12, 1820, 4, 12], [13, 560, 3, 13], [14, 120, 2, 14], [15, 16, 1, 15], [16, 1, 0, 16]], complete_k_zero_through_n_coefficients).

(17, [[0, 1, 17, 0], [1, 17, 16, 1], [2, 136, 15, 2], [3, 680, 14, 3], [4, 2380, 13, 4], [5, 6188, 12, 5], [6, 12376, 11, 6], [7, 19448, 10, 7], [8, 24310, 9, 8], [9, 24310, 8, 9], [10, 19448, 7, 10], [11, 12376, 6, 11], [12, 6188, 5, 12], [13, 2380, 4, 13], [14, 680, 3, 14], [15, 136, 2, 15], [16, 17, 1, 16], [17, 1, 0, 17]], complete_k_zero_through_n_coefficients)

Every row is [k,C(n,k),x exponent n−k,y exponent k]. Retain all n+1 rows and exact integer coefficients. At n=0 the expansion is constant1, including the formal zero-exponent convention. Noncommuting variables require a different theorem. The exact result, retaining the requested order, is (17, [[0, 1, 17, 0], [1, 17, 16, 1], [2, 136, 15, 2], [3, 680, 14, 3], [4, 2380, 13, 4], [5, 6188, 12, 5], [6, 12376, 11, 6], [7, 19448, 10, 7], [8, 24310, 9, 8], [9, 24310, 8, 9], [10, 19448, 7, 10], [11, 12376, 6, 11], [12, 6188, 5, 12], [13, 2380, 4, 13], [14, 680, 3, 14], [15, 136, 2, 15], [16, 17, 1, 16], [17, 1, 0, 17]], complete_k_zero_through_n_coefficients).

(18, [[0, 1, 18, 0], [1, 18, 17, 1], [2, 153, 16, 2], [3, 816, 15, 3], [4, 3060, 14, 4], [5, 8568, 13, 5], [6, 18564, 12, 6], [7, 31824, 11, 7], [8, 43758, 10, 8], [9, 48620, 9, 9], [10, 43758, 8, 10], [11, 31824, 7, 11], [12, 18564, 6, 12], [13, 8568, 5, 13], [14, 3060, 4, 14], [15, 816, 3, 15], [16, 153, 2, 16], [17, 18, 1, 17], [18, 1, 0, 18]], complete_k_zero_through_n_coefficients)

Every row is [k,C(n,k),x exponent n−k,y exponent k]. Retain all n+1 rows and exact integer coefficients. At n=0 the expansion is constant1, including the formal zero-exponent convention. Noncommuting variables require a different theorem. The exact result, retaining the requested order, is (18, [[0, 1, 18, 0], [1, 18, 17, 1], [2, 153, 16, 2], [3, 816, 15, 3], [4, 3060, 14, 4], [5, 8568, 13, 5], [6, 18564, 12, 6], [7, 31824, 11, 7], [8, 43758, 10, 8], [9, 48620, 9, 9], [10, 43758, 8, 10], [11, 31824, 7, 11], [12, 18564, 6, 12], [13, 8568, 5, 13], [14, 3060, 4, 14], [15, 816, 3, 15], [16, 153, 2, 16], [17, 18, 1, 17], [18, 1, 0, 18]], complete_k_zero_through_n_coefficients).

(19, [[0, 1, 19, 0], [1, 19, 18, 1], [2, 171, 17, 2], [3, 969, 16, 3], [4, 3876, 15, 4], [5, 11628, 14, 5], [6, 27132, 13, 6], [7, 50388, 12, 7], [8, 75582, 11, 8], [9, 92378, 10, 9], [10, 92378, 9, 10], [11, 75582, 8, 11], [12, 50388, 7, 12], [13, 27132, 6, 13], [14, 11628, 5, 14], [15, 3876, 4, 15], [16, 969, 3, 16], [17, 171, 2, 17], [18, 19, 1, 18], [19, 1, 0, 19]], complete_k_zero_through_n_coefficients)

Every row is [k,C(n,k),x exponent n−k,y exponent k]. Retain all n+1 rows and exact integer coefficients. At n=0 the expansion is constant1, including the formal zero-exponent convention. Noncommuting variables require a different theorem. The exact result, retaining the requested order, is (19, [[0, 1, 19, 0], [1, 19, 18, 1], [2, 171, 17, 2], [3, 969, 16, 3], [4, 3876, 15, 4], [5, 11628, 14, 5], [6, 27132, 13, 6], [7, 50388, 12, 7], [8, 75582, 11, 8], [9, 92378, 10, 9], [10, 92378, 9, 10], [11, 75582, 8, 11], [12, 50388, 7, 12], [13, 27132, 6, 13], [14, 11628, 5, 14], [15, 3876, 4, 15], [16, 969, 3, 16], [17, 171, 2, 17], [18, 19, 1, 18], [19, 1, 0, 19]], complete_k_zero_through_n_coefficients).

(20, [[0, 1, 20, 0], [1, 20, 19, 1], [2, 190, 18, 2], [3, 1140, 17, 3], [4, 4845, 16, 4], [5, 15504, 15, 5], [6, 38760, 14, 6], [7, 77520, 13, 7], [8, 125970, 12, 8], [9, 167960, 11, 9], [10, 184756, 10, 10], [11, 167960, 9, 11], [12, 125970, 8, 12], [13, 77520, 7, 13], [14, 38760, 6, 14], [15, 15504, 5, 15], [16, 4845, 4, 16], [17, 1140, 3, 17], [18, 190, 2, 18], [19, 20, 1, 19], [20, 1, 0, 20]], complete_k_zero_through_n_coefficients)

Every row is [k,C(n,k),x exponent n−k,y exponent k]. Retain all n+1 rows and exact integer coefficients. At n=0 the expansion is constant1, including the formal zero-exponent convention. Noncommuting variables require a different theorem. The exact result, retaining the requested order, is (20, [[0, 1, 20, 0], [1, 20, 19, 1], [2, 190, 18, 2], [3, 1140, 17, 3], [4, 4845, 16, 4], [5, 15504, 15, 5], [6, 38760, 14, 6], [7, 77520, 13, 7], [8, 125970, 12, 8], [9, 167960, 11, 9], [10, 184756, 10, 10], [11, 167960, 9, 11], [12, 125970, 8, 12], [13, 77520, 7, 13], [14, 38760, 6, 14], [15, 15504, 5, 15], [16, 4845, 4, 16], [17, 1140, 3, 17], [18, 190, 2, 18], [19, 20, 1, 19], [20, 1, 0, 20]], complete_k_zero_through_n_coefficients).

(21, [[0, 1, 21, 0], [1, 21, 20, 1], [2, 210, 19, 2], [3, 1330, 18, 3], [4, 5985, 17, 4], [5, 20349, 16, 5], [6, 54264, 15, 6], [7, 116280, 14, 7], [8, 203490, 13, 8], [9, 293930, 12, 9], [10, 352716, 11, 10], [11, 352716, 10, 11], [12, 293930, 9, 12], [13, 203490, 8, 13], [14, 116280, 7, 14], [15, 54264, 6, 15], [16, 20349, 5, 16], [17, 5985, 4, 17], [18, 1330, 3, 18], [19, 210, 2, 19], [20, 21, 1, 20], [21, 1, 0, 21]], complete_k_zero_through_n_coefficients)

Every row is [k,C(n,k),x exponent n−k,y exponent k]. Retain all n+1 rows and exact integer coefficients. At n=0 the expansion is constant1, including the formal zero-exponent convention. Noncommuting variables require a different theorem. The exact result, retaining the requested order, is (21, [[0, 1, 21, 0], [1, 21, 20, 1], [2, 210, 19, 2], [3, 1330, 18, 3], [4, 5985, 17, 4], [5, 20349, 16, 5], [6, 54264, 15, 6], [7, 116280, 14, 7], [8, 203490, 13, 8], [9, 293930, 12, 9], [10, 352716, 11, 10], [11, 352716, 10, 11], [12, 293930, 9, 12], [13, 203490, 8, 13], [14, 116280, 7, 14], [15, 54264, 6, 15], [16, 20349, 5, 16], [17, 5985, 4, 17], [18, 1330, 3, 18], [19, 210, 2, 19], [20, 21, 1, 20], [21, 1, 0, 21]], complete_k_zero_through_n_coefficients).

(22, [[0, 1, 22, 0], [1, 22, 21, 1], [2, 231, 20, 2], [3, 1540, 19, 3], [4, 7315, 18, 4], [5, 26334, 17, 5], [6, 74613, 16, 6], [7, 170544, 15, 7], [8, 319770, 14, 8], [9, 497420, 13, 9], [10, 646646, 12, 10], [11, 705432, 11, 11], [12, 646646, 10, 12], [13, 497420, 9, 13], [14, 319770, 8, 14], [15, 170544, 7, 15], [16, 74613, 6, 16], [17, 26334, 5, 17], [18, 7315, 4, 18], [19, 1540, 3, 19], [20, 231, 2, 20], [21, 22, 1, 21], [22, 1, 0, 22]], complete_k_zero_through_n_coefficients)

Every row is [k,C(n,k),x exponent n−k,y exponent k]. Retain all n+1 rows and exact integer coefficients. At n=0 the expansion is constant1, including the formal zero-exponent convention. Noncommuting variables require a different theorem. The exact result, retaining the requested order, is (22, [[0, 1, 22, 0], [1, 22, 21, 1], [2, 231, 20, 2], [3, 1540, 19, 3], [4, 7315, 18, 4], [5, 26334, 17, 5], [6, 74613, 16, 6], [7, 170544, 15, 7], [8, 319770, 14, 8], [9, 497420, 13, 9], [10, 646646, 12, 10], [11, 705432, 11, 11], [12, 646646, 10, 12], [13, 497420, 9, 13], [14, 319770, 8, 14], [15, 170544, 7, 15], [16, 74613, 6, 16], [17, 26334, 5, 17], [18, 7315, 4, 18], [19, 1540, 3, 19], [20, 231, 2, 20], [21, 22, 1, 21], [22, 1, 0, 22]], complete_k_zero_through_n_coefficients).

(23, [[0, 1, 23, 0], [1, 23, 22, 1], [2, 253, 21, 2], [3, 1771, 20, 3], [4, 8855, 19, 4], [5, 33649, 18, 5], [6, 100947, 17, 6], [7, 245157, 16, 7], [8, 490314, 15, 8], [9, 817190, 14, 9], [10, 1144066, 13, 10], [11, 1352078, 12, 11], [12, 1352078, 11, 12], [13, 1144066, 10, 13], [14, 817190, 9, 14], [15, 490314, 8, 15], [16, 245157, 7, 16], [17, 100947, 6, 17], [18, 33649, 5, 18], [19, 8855, 4, 19], [20, 1771, 3, 20], [21, 253, 2, 21], [22, 23, 1, 22], [23, 1, 0, 23]], complete_k_zero_through_n_coefficients)

Every row is [k,C(n,k),x exponent n−k,y exponent k]. Retain all n+1 rows and exact integer coefficients. At n=0 the expansion is constant1, including the formal zero-exponent convention. Noncommuting variables require a different theorem. The exact result, retaining the requested order, is (23, [[0, 1, 23, 0], [1, 23, 22, 1], [2, 253, 21, 2], [3, 1771, 20, 3], [4, 8855, 19, 4], [5, 33649, 18, 5], [6, 100947, 17, 6], [7, 245157, 16, 7], [8, 490314, 15, 8], [9, 817190, 14, 9], [10, 1144066, 13, 10], [11, 1352078, 12, 11], [12, 1352078, 11, 12], [13, 1144066, 10, 13], [14, 817190, 9, 14], [15, 490314, 8, 15], [16, 245157, 7, 16], [17, 100947, 6, 17], [18, 33649, 5, 18], [19, 8855, 4, 19], [20, 1771, 3, 20], [21, 253, 2, 21], [22, 23, 1, 22], [23, 1, 0, 23]], complete_k_zero_through_n_coefficients).

(24, [[0, 1, 24, 0], [1, 24, 23, 1], [2, 276, 22, 2], [3, 2024, 21, 3], [4, 10626, 20, 4], [5, 42504, 19, 5], [6, 134596, 18, 6], [7, 346104, 17, 7], [8, 735471, 16, 8], [9, 1307504, 15, 9], [10, 1961256, 14, 10], [11, 2496144, 13, 11], [12, 2704156, 12, 12], [13, 2496144, 11, 13], [14, 1961256, 10, 14], [15, 1307504, 9, 15], [16, 735471, 8, 16], [17, 346104, 7, 17], [18, 134596, 6, 18], [19, 42504, 5, 19], [20, 10626, 4, 20], [21, 2024, 3, 21], [22, 276, 2, 22], [23, 24, 1, 23], [24, 1, 0, 24]], complete_k_zero_through_n_coefficients)

Every row is [k,C(n,k),x exponent n−k,y exponent k]. Retain all n+1 rows and exact integer coefficients. At n=0 the expansion is constant1, including the formal zero-exponent convention. Noncommuting variables require a different theorem. The exact result, retaining the requested order, is (24, [[0, 1, 24, 0], [1, 24, 23, 1], [2, 276, 22, 2], [3, 2024, 21, 3], [4, 10626, 20, 4], [5, 42504, 19, 5], [6, 134596, 18, 6], [7, 346104, 17, 7], [8, 735471, 16, 8], [9, 1307504, 15, 9], [10, 1961256, 14, 10], [11, 2496144, 13, 11], [12, 2704156, 12, 12], [13, 2496144, 11, 13], [14, 1961256, 10, 14], [15, 1307504, 9, 15], [16, 735471, 8, 16], [17, 346104, 7, 17], [18, 134596, 6, 18], [19, 42504, 5, 19], [20, 10626, 4, 20], [21, 2024, 3, 21], [22, 276, 2, 22], [23, 24, 1, 23], [24, 1, 0, 24]], complete_k_zero_through_n_coefficients).

(0, -3, 2, 1, 1, Yes)

Evaluate (x+y)^n and the sum of C(n,k)*x^(n−k)*y^k over every k=0..n with exact BigInt arithmetic. The polynomial convention uses zeroth power1, including a zero base. No floating-point approximation or noncommutative identity is inferred. The exact result, retaining the requested order, is (0, -3, 2, 1, 1, Yes).

(1, -2, 1, -1, -1, Yes)

Evaluate (x+y)^n and the sum of C(n,k)*x^(n−k)*y^k over every k=0..n with exact BigInt arithmetic. The polynomial convention uses zeroth power1, including a zero base. No floating-point approximation or noncommutative identity is inferred. The exact result, retaining the requested order, is (1, -2, 1, -1, -1, Yes).

(2, -1, 0, 1, 1, Yes)

Evaluate (x+y)^n and the sum of C(n,k)*x^(n−k)*y^k over every k=0..n with exact BigInt arithmetic. The polynomial convention uses zeroth power1, including a zero base. No floating-point approximation or noncommutative identity is inferred. The exact result, retaining the requested order, is (2, -1, 0, 1, 1, Yes).

(3, 0, -1, -1, -1, Yes)

Evaluate (x+y)^n and the sum of C(n,k)*x^(n−k)*y^k over every k=0..n with exact BigInt arithmetic. The polynomial convention uses zeroth power1, including a zero base. No floating-point approximation or noncommutative identity is inferred. The exact result, retaining the requested order, is (3, 0, -1, -1, -1, Yes).

(4, 1, -2, 1, 1, Yes)

Evaluate (x+y)^n and the sum of C(n,k)*x^(n−k)*y^k over every k=0..n with exact BigInt arithmetic. The polynomial convention uses zeroth power1, including a zero base. No floating-point approximation or noncommutative identity is inferred. The exact result, retaining the requested order, is (4, 1, -2, 1, 1, Yes).

(5, 2, 2, 1024, 1024, Yes)

Evaluate (x+y)^n and the sum of C(n,k)*x^(n−k)*y^k over every k=0..n with exact BigInt arithmetic. The polynomial convention uses zeroth power1, including a zero base. No floating-point approximation or noncommutative identity is inferred. The exact result, retaining the requested order, is (5, 2, 2, 1024, 1024, Yes).

(6, 3, 1, 4096, 4096, Yes)

Evaluate (x+y)^n and the sum of C(n,k)*x^(n−k)*y^k over every k=0..n with exact BigInt arithmetic. The polynomial convention uses zeroth power1, including a zero base. No floating-point approximation or noncommutative identity is inferred. The exact result, retaining the requested order, is (6, 3, 1, 4096, 4096, Yes).

(7, -3, 0, -2187, -2187, Yes)

Evaluate (x+y)^n and the sum of C(n,k)*x^(n−k)*y^k over every k=0..n with exact BigInt arithmetic. The polynomial convention uses zeroth power1, including a zero base. No floating-point approximation or noncommutative identity is inferred. The exact result, retaining the requested order, is (7, -3, 0, -2187, -2187, Yes).

(8, -2, -1, 6561, 6561, Yes)

Evaluate (x+y)^n and the sum of C(n,k)*x^(n−k)*y^k over every k=0..n with exact BigInt arithmetic. The polynomial convention uses zeroth power1, including a zero base. No floating-point approximation or noncommutative identity is inferred. The exact result, retaining the requested order, is (8, -2, -1, 6561, 6561, Yes).

(9, -1, -2, -19683, -19683, Yes)

Evaluate (x+y)^n and the sum of C(n,k)*x^(n−k)*y^k over every k=0..n with exact BigInt arithmetic. The polynomial convention uses zeroth power1, including a zero base. No floating-point approximation or noncommutative identity is inferred. The exact result, retaining the requested order, is (9, -1, -2, -19683, -19683, Yes).

(10, 0, 2, 1024, 1024, Yes)

Evaluate (x+y)^n and the sum of C(n,k)*x^(n−k)*y^k over every k=0..n with exact BigInt arithmetic. The polynomial convention uses zeroth power1, including a zero base. No floating-point approximation or noncommutative identity is inferred. The exact result, retaining the requested order, is (10, 0, 2, 1024, 1024, Yes).

(11, 1, 1, 2048, 2048, Yes)

Evaluate (x+y)^n and the sum of C(n,k)*x^(n−k)*y^k over every k=0..n with exact BigInt arithmetic. The polynomial convention uses zeroth power1, including a zero base. No floating-point approximation or noncommutative identity is inferred. The exact result, retaining the requested order, is (11, 1, 1, 2048, 2048, Yes).

(12, 2, 0, 4096, 4096, Yes)

Evaluate (x+y)^n and the sum of C(n,k)*x^(n−k)*y^k over every k=0..n with exact BigInt arithmetic. The polynomial convention uses zeroth power1, including a zero base. No floating-point approximation or noncommutative identity is inferred. The exact result, retaining the requested order, is (12, 2, 0, 4096, 4096, Yes).

(13, 3, -1, 8192, 8192, Yes)

Evaluate (x+y)^n and the sum of C(n,k)*x^(n−k)*y^k over every k=0..n with exact BigInt arithmetic. The polynomial convention uses zeroth power1, including a zero base. No floating-point approximation or noncommutative identity is inferred. The exact result, retaining the requested order, is (13, 3, -1, 8192, 8192, Yes).

(14, -3, -2, 6103515625, 6103515625, Yes)

Evaluate (x+y)^n and the sum of C(n,k)*x^(n−k)*y^k over every k=0..n with exact BigInt arithmetic. The polynomial convention uses zeroth power1, including a zero base. No floating-point approximation or noncommutative identity is inferred. The exact result, retaining the requested order, is (14, -3, -2, 6103515625, 6103515625, Yes).

(15, -2, 2, 0, 0, Yes)

Evaluate (x+y)^n and the sum of C(n,k)*x^(n−k)*y^k over every k=0..n with exact BigInt arithmetic. The polynomial convention uses zeroth power1, including a zero base. No floating-point approximation or noncommutative identity is inferred. The exact result, retaining the requested order, is (15, -2, 2, 0, 0, Yes).

(16, -1, 1, 0, 0, Yes)

Evaluate (x+y)^n and the sum of C(n,k)*x^(n−k)*y^k over every k=0..n with exact BigInt arithmetic. The polynomial convention uses zeroth power1, including a zero base. No floating-point approximation or noncommutative identity is inferred. The exact result, retaining the requested order, is (16, -1, 1, 0, 0, Yes).

(17, 0, 0, 0, 0, Yes)

Evaluate (x+y)^n and the sum of C(n,k)*x^(n−k)*y^k over every k=0..n with exact BigInt arithmetic. The polynomial convention uses zeroth power1, including a zero base. No floating-point approximation or noncommutative identity is inferred. The exact result, retaining the requested order, is (17, 0, 0, 0, 0, Yes).

(18, 1, -1, 0, 0, Yes)

Evaluate (x+y)^n and the sum of C(n,k)*x^(n−k)*y^k over every k=0..n with exact BigInt arithmetic. The polynomial convention uses zeroth power1, including a zero base. No floating-point approximation or noncommutative identity is inferred. The exact result, retaining the requested order, is (18, 1, -1, 0, 0, Yes).

(19, 2, -2, 0, 0, Yes)

Evaluate (x+y)^n and the sum of C(n,k)*x^(n−k)*y^k over every k=0..n with exact BigInt arithmetic. The polynomial convention uses zeroth power1, including a zero base. No floating-point approximation or noncommutative identity is inferred. The exact result, retaining the requested order, is (19, 2, -2, 0, 0, Yes).

(20, 3, 2, 95367431640625, 95367431640625, Yes)

Evaluate (x+y)^n and the sum of C(n,k)*x^(n−k)*y^k over every k=0..n with exact BigInt arithmetic. The polynomial convention uses zeroth power1, including a zero base. No floating-point approximation or noncommutative identity is inferred. The exact result, retaining the requested order, is (20, 3, 2, 95367431640625, 95367431640625, Yes).

(21, -3, 1, -2097152, -2097152, Yes)

Evaluate (x+y)^n and the sum of C(n,k)*x^(n−k)*y^k over every k=0..n with exact BigInt arithmetic. The polynomial convention uses zeroth power1, including a zero base. No floating-point approximation or noncommutative identity is inferred. The exact result, retaining the requested order, is (21, -3, 1, -2097152, -2097152, Yes).

(22, -2, 0, 4194304, 4194304, Yes)

Evaluate (x+y)^n and the sum of C(n,k)*x^(n−k)*y^k over every k=0..n with exact BigInt arithmetic. The polynomial convention uses zeroth power1, including a zero base. No floating-point approximation or noncommutative identity is inferred. The exact result, retaining the requested order, is (22, -2, 0, 4194304, 4194304, Yes).

(23, -1, -1, -8388608, -8388608, Yes)

Evaluate (x+y)^n and the sum of C(n,k)*x^(n−k)*y^k over every k=0..n with exact BigInt arithmetic. The polynomial convention uses zeroth power1, including a zero base. No floating-point approximation or noncommutative identity is inferred. The exact result, retaining the requested order, is (23, -1, -1, -8388608, -8388608, Yes).

(24, 0, -2, 16777216, 16777216, Yes)

Evaluate (x+y)^n and the sum of C(n,k)*x^(n−k)*y^k over every k=0..n with exact BigInt arithmetic. The polynomial convention uses zeroth power1, including a zero base. No floating-point approximation or noncommutative identity is inferred. The exact result, retaining the requested order, is (24, 0, -2, 16777216, 16777216, Yes).

(-1, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)

The fixed No withholds the natural-exponent finite binomial-polynomial claim. It is not a statement that other series expansions cannot exist; those need separate hypotheses and convergence conventions. The exact result, retaining the requested order, is (-1, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication).

(-2, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)

The fixed No withholds the natural-exponent finite binomial-polynomial claim. It is not a statement that other series expansions cannot exist; those need separate hypotheses and convergence conventions. The exact result, retaining the requested order, is (-2, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication).

(-3, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)

The fixed No withholds the natural-exponent finite binomial-polynomial claim. It is not a statement that other series expansions cannot exist; those need separate hypotheses and convergence conventions. The exact result, retaining the requested order, is (-3, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication).

(-4, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)

The fixed No withholds the natural-exponent finite binomial-polynomial claim. It is not a statement that other series expansions cannot exist; those need separate hypotheses and convergence conventions. The exact result, retaining the requested order, is (-4, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication).

(-5, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)

The fixed No withholds the natural-exponent finite binomial-polynomial claim. It is not a statement that other series expansions cannot exist; those need separate hypotheses and convergence conventions. The exact result, retaining the requested order, is (-5, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication).

(-6, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)

The fixed No withholds the natural-exponent finite binomial-polynomial claim. It is not a statement that other series expansions cannot exist; those need separate hypotheses and convergence conventions. The exact result, retaining the requested order, is (-6, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication).

(-7, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)

The fixed No withholds the natural-exponent finite binomial-polynomial claim. It is not a statement that other series expansions cannot exist; those need separate hypotheses and convergence conventions. The exact result, retaining the requested order, is (-7, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication).

(-8, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)

The fixed No withholds the natural-exponent finite binomial-polynomial claim. It is not a statement that other series expansions cannot exist; those need separate hypotheses and convergence conventions. The exact result, retaining the requested order, is (-8, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication).

(-9, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)

The fixed No withholds the natural-exponent finite binomial-polynomial claim. It is not a statement that other series expansions cannot exist; those need separate hypotheses and convergence conventions. The exact result, retaining the requested order, is (-9, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication).

(-10, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)

The fixed No withholds the natural-exponent finite binomial-polynomial claim. It is not a statement that other series expansions cannot exist; those need separate hypotheses and convergence conventions. The exact result, retaining the requested order, is (-10, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication).

(-11, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)

The fixed No withholds the natural-exponent finite binomial-polynomial claim. It is not a statement that other series expansions cannot exist; those need separate hypotheses and convergence conventions. The exact result, retaining the requested order, is (-11, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication).

(-12, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)

The fixed No withholds the natural-exponent finite binomial-polynomial claim. It is not a statement that other series expansions cannot exist; those need separate hypotheses and convergence conventions. The exact result, retaining the requested order, is (-12, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication).

(-13, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)

The fixed No withholds the natural-exponent finite binomial-polynomial claim. It is not a statement that other series expansions cannot exist; those need separate hypotheses and convergence conventions. The exact result, retaining the requested order, is (-13, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication).

(-14, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)

The fixed No withholds the natural-exponent finite binomial-polynomial claim. It is not a statement that other series expansions cannot exist; those need separate hypotheses and convergence conventions. The exact result, retaining the requested order, is (-14, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication).

(-15, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)

The fixed No withholds the natural-exponent finite binomial-polynomial claim. It is not a statement that other series expansions cannot exist; those need separate hypotheses and convergence conventions. The exact result, retaining the requested order, is (-15, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication).

(-16, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)

The fixed No withholds the natural-exponent finite binomial-polynomial claim. It is not a statement that other series expansions cannot exist; those need separate hypotheses and convergence conventions. The exact result, retaining the requested order, is (-16, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication).

(-17, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)

The fixed No withholds the natural-exponent finite binomial-polynomial claim. It is not a statement that other series expansions cannot exist; those need separate hypotheses and convergence conventions. The exact result, retaining the requested order, is (-17, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication).

(-18, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)

The fixed No withholds the natural-exponent finite binomial-polynomial claim. It is not a statement that other series expansions cannot exist; those need separate hypotheses and convergence conventions. The exact result, retaining the requested order, is (-18, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication).

(-19, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)

The fixed No withholds the natural-exponent finite binomial-polynomial claim. It is not a statement that other series expansions cannot exist; those need separate hypotheses and convergence conventions. The exact result, retaining the requested order, is (-19, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication).

(-20, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)

The fixed No withholds the natural-exponent finite binomial-polynomial claim. It is not a statement that other series expansions cannot exist; those need separate hypotheses and convergence conventions. The exact result, retaining the requested order, is (-20, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication).

(-21, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)

The fixed No withholds the natural-exponent finite binomial-polynomial claim. It is not a statement that other series expansions cannot exist; those need separate hypotheses and convergence conventions. The exact result, retaining the requested order, is (-21, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication).

(-22, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)

The fixed No withholds the natural-exponent finite binomial-polynomial claim. It is not a statement that other series expansions cannot exist; those need separate hypotheses and convergence conventions. The exact result, retaining the requested order, is (-22, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication).

(-23, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)

The fixed No withholds the natural-exponent finite binomial-polynomial claim. It is not a statement that other series expansions cannot exist; those need separate hypotheses and convergence conventions. The exact result, retaining the requested order, is (-23, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication).

(-24, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)

The fixed No withholds the natural-exponent finite binomial-polynomial claim. It is not a statement that other series expansions cannot exist; those need separate hypotheses and convergence conventions. The exact result, retaining the requested order, is (-24, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication).

(-25, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)

The fixed No withholds the natural-exponent finite binomial-polynomial claim. It is not a statement that other series expansions cannot exist; those need separate hypotheses and convergence conventions. The exact result, retaining the requested order, is (-25, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication).