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, recurrence_value, closed_value, difference_value, all_equal, certificate).
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, recurrence_value, closed_value, difference_value, all_equal, 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.
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.
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.
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.
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.