Original question and givens
Use natural index n=0≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, central_binomial, denominator, remainder, quotient, binomial_difference, equal).
G001 — Use natural index n=0≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, central_binomial, denominator, remainder, quotient, binomial_difference, equal).
Step-by-step derivation
S001 · implies
Before
source operation=catalan-closed-form;original snapshot and mathematics retained
After
explicit ordered source output fields=[n, central_binomial, denominator, remainder, quotient, binomial_difference, equal]
Conditions: none
Keep the historical interface marker and list ordered source fields without JavaScript.Tags are not additional evidence;fixed No markers retain their scope.The ordering helper supplies protocol order,not mathematical correctness.
Step identifiers and dependencies
Rule: source_ordered_interface_explicit_without_javascript · Uses: G001
S002 · implies
Before
ORIGINAL requested natural index n=0
After
initialize C_0=1;recurrence C_(j+1)=sum_i=0..j C_i*C_(j-i),including ALL splits
Conditions: none
There is one empty object,so C0=1.Include every split i0..j for C_(j+1),without a C1-based index shift.
Step identifiers and dependencies
Rule: catalan_base_case_empty_object_C0_one · Uses: G001, S001
S003 · implies
Before
Catalan central:ORIGINAL binomial N=0,K=0
After
use k=min(K,N-K)=0;initialize exact product1;empty product for k0 is1
Conditions: none
Use binomial symmetry to shorten the product while retaining original N/K roles.Multiply before each exact division,without floating-point factorial approximations.
Step identifiers and dependencies
Rule: catalan_exact_binomial_product_initial_and_symmetry · Uses: G001, S002
S004 · calculation
Before
Catalan ORIGINAL denominator n plus1 := sum(0, 1)
After
Catalan ORIGINAL denominator n plus1 = 1
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S003
S005 · implies
Before
ORIGINAL n=0;central binomial=1;positive denominator n+1=1
After
Euclidean quotient=1,remainder=0;check remainder0 BEFORE treating quotient as Catalan integer
Conditions: none
Check remainder0 before taking an exact integer quotient.Do not hide nondivisibility by truncation;retain denominator1 and central1 at n0.
Step identifiers and dependencies
Rule: catalan_remainder_BEFORE_exact_closed_division · Uses: G001, S004
S006 · calculation
Before
Catalan exact closed quotient after remainder0 := div(1, 1)
After
Catalan exact closed quotient after remainder0 = 1
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S005
S007 · implies
Before
Catalan adjacent:N=0,K=1
After
K outside0..N,so binomial=0
Conditions: none
The source convention gives0 outside0..N,including choose(0,1) when n0.
Step identifiers and dependencies
Rule: catalan_binomial_out_of_range_zero · Uses: G001, S006
S008 · calculation
Before
Catalan central minus adjacent binomial := sub(1, 0)
After
Catalan central minus adjacent binomial = 1
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S007
S009 · implies
Before
ORIGINAL n=0;recurrence value=1;central=1,adjacent=0,denominator=1
After
closed quotient=1;binomial difference=1;ALL THREE agree;balanced paths counted by choose(2n,n),bad paths first-crossing reflection counted by choose(2n,n+1)
Conditions: none
Reflect a bad balanced path through its first crossing of−1 to obtain a path with n+1 ups and n−1 downs,so good paths are the binomial difference.First-return decomposition U P D Q gives the Catalan convolution with empty-path base1,uniquely determining the same count.The adjacent-binomial ratio yields central/(n+1).This is a general argument,not extrapolation from finitely many equalities.
Step identifiers and dependencies
Rule: catalan_general_Dyck_reflection_and_first_return_recurrence · Uses: G001, S008