Question 24012: Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial

Original question and givens

Use the derivative values=[12, 24, 36, 48, 60, 72] at the fixed center=-1, through declared order=5≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, shifted_coefficients, indexed_coefficients, certificate).

• G001 — Use the derivative values=[12, 24, 36, 48, 60, 72] at the fixed center=-1, through declared order=5≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, shifted_coefficients, indexed_coefficients, certificate).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-taylor-polynomial

Step-by-step derivation

source roles = [taylor-coefficient-construction]

derived = [[center, order, shifted_coefficients, indexed_coefficients, certificate]]

source roles = [-1, 5, [12, 24, 36, 48, 60, 72]]

derived = [[0, 1, 2, 3, 4, 5], 6, Yes]

source roles = [0]

derived = [1]

Taylor coefficient0 derivative over factorial := div(12, 1)

Taylor coefficient0 derivative over factorial = 12

Taylor k1 factorial step1 := mul(1, 1)

Taylor k1 factorial step1 = 1

Taylor coefficient1 derivative over factorial := div(24, 1)

Taylor coefficient1 derivative over factorial = 24

Taylor k2 factorial step1 := mul(1, 1)

Taylor k2 factorial step1 = 1

Taylor k2 factorial step2 := mul(1, 2)

Taylor k2 factorial step2 = 2

Taylor coefficient2 derivative over factorial := div(36, 2)

Taylor coefficient2 derivative over factorial = 18

Taylor k3 factorial step1 := mul(1, 1)

Taylor k3 factorial step1 = 1

Taylor k3 factorial step2 := mul(1, 2)

Taylor k3 factorial step2 = 2

Taylor k3 factorial step3 := mul(2, 3)

Taylor k3 factorial step3 = 6

Taylor coefficient3 derivative over factorial := div(48, 6)

Taylor coefficient3 derivative over factorial = 8

Taylor k4 factorial step1 := mul(1, 1)

Taylor k4 factorial step1 = 1

Taylor k4 factorial step2 := mul(1, 2)

Taylor k4 factorial step2 = 2

Taylor k4 factorial step3 := mul(2, 3)

Taylor k4 factorial step3 = 6

Taylor k4 factorial step4 := mul(6, 4)

Taylor k4 factorial step4 = 24

Taylor coefficient4 derivative over factorial := div(60, 24)

Taylor coefficient4 derivative over factorial = 5/2

Taylor k5 factorial step1 := mul(1, 1)

Taylor k5 factorial step1 = 1

Taylor k5 factorial step2 := mul(1, 2)

Taylor k5 factorial step2 = 2

Taylor k5 factorial step3 := mul(2, 3)

Taylor k5 factorial step3 = 6

Taylor k5 factorial step4 := mul(6, 4)

Taylor k5 factorial step4 = 24

Taylor k5 factorial step5 := mul(24, 5)

Taylor k5 factorial step5 = 120

Taylor coefficient5 derivative over factorial := div(72, 120)

Taylor coefficient5 derivative over factorial = 3/5

source roles = [-1, 5, [12, 24, 18, 8, 5/2, 3/5]]

derived = [T_n(x)=sum c_k*(x-center)^k, T_n^(j)(center)=declared derivative[j] for EVERY0<=j<=n, no remainder or infinite-series certificate]

Reverse checks

• S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S024 · premise_and_result_audit — Recompute Taylor coefficient5 derivative over factorial with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute Taylor k5 factorial step5 with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute Taylor k5 factorial step4 with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute Taylor k5 factorial step3 with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute Taylor k5 factorial step2 with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute Taylor k5 factorial step1 with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute Taylor coefficient4 derivative over factorial with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute Taylor k4 factorial step4 with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute Taylor k4 factorial step3 with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute Taylor k4 factorial step2 with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute Taylor k4 factorial step1 with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute Taylor coefficient3 derivative over factorial with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute Taylor k3 factorial step3 with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute Taylor k3 factorial step2 with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute Taylor k3 factorial step1 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute Taylor coefficient2 derivative over factorial with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute Taylor k2 factorial step2 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute Taylor k2 factorial step1 with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute Taylor coefficient1 derivative over factorial with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute Taylor k1 factorial step1 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Taylor coefficient0 derivative over factorial with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(-1, 5, [12, 24, 18, 8, 5/2, 3/5], [[0, 12], [1, 24], [2, 18], [3, 8], [4, 5/2], [5, 3/5]], coefficient_k_equals_derivative_k_over_k_factorial)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.