第 24633 題：解析普通生成函數

原題與已知條件

A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-finite-partial-sum；資料={"finite_support_coefficients":[8,16,24,32,40,48],"x":[1,3],"cutoff":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-finite-partial-sum；資料={"finite_support_coefficients":[8,16,24,32,40,48],"x":[1,3],"cutoff":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

定義與適用範圍

沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。

• mko-analytic-ordinary-generating-function

逐步推導

source roles = [ogf-finite-partial-sum]

derived = [[finite_support_coefficients, x, cutoff, term_rows, finite_sum, complete_support_included, certificate]]

source roles = [[8, 16, 24, 32, 40, 48]]

derived = [[8, 16, 24, 32, 40, 48], a_n=0 for EVERY n beyond this finite list]

(1/3)^0

OGF ORIGINAL x power0 = 1

OGF term0 coefficient times power := mul(8, 1)

OGF term0 coefficient times power = 8

OGF partial sum through0 := sum(0, 8)

OGF partial sum through0 = 8

(1/3)^1

OGF ORIGINAL x power1 = 1/3

OGF term1 coefficient times power := mul(16, 1/3)

OGF term1 coefficient times power = 16/3

OGF partial sum through1 := sum(8, 16/3)

OGF partial sum through1 = 40/3

(1/3)^2

OGF ORIGINAL x power2 = 1/9

OGF term2 coefficient times power := mul(24, 1/9)

OGF term2 coefficient times power = 8/3

OGF partial sum through2 := sum(40/3, 8/3)

OGF partial sum through2 = 16

(1/3)^3

OGF ORIGINAL x power3 = 1/27

OGF term3 coefficient times power := mul(32, 1/27)

OGF term3 coefficient times power = 32/27

OGF partial sum through3 := sum(16, 32/27)

OGF partial sum through3 = 464/27

(1/3)^4

OGF ORIGINAL x power4 = 1/81

OGF term4 coefficient times power := mul(40, 1/81)

OGF term4 coefficient times power = 40/81

OGF partial sum through4 := sum(464/27, 40/81)

OGF partial sum through4 = 1432/81

source roles = [[8, 16, 24, 32, 40, 48], 1/3, 4]

derived = [1432/81, 否]

逐步反向核對

• S018 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S017 · premise_and_result_audit — 以精確有理數重算 OGF partial sum through4，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 OGF term4 coefficient times power，不以小數近似替代。
• S015 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S014 · premise_and_result_audit — 以精確有理數重算 OGF partial sum through3，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 OGF term3 coefficient times power，不以小數近似替代。
• S012 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S011 · premise_and_result_audit — 以精確有理數重算 OGF partial sum through2，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 OGF term2 coefficient times power，不以小數近似替代。
• S009 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S008 · premise_and_result_audit — 以精確有理數重算 OGF partial sum through1，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 OGF term1 coefficient times power，不以小數近似替代。
• S006 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S005 · premise_and_result_audit — 以精確有理數重算 OGF partial sum through0，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 OGF term0 coefficient times power，不以小數近似替代。
• S003 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S002 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S001 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。

結論

([8, 16, 24, 32, 40, 48], 1/3, 4, [[0, 8, 8], [1, 16, 16/3], [2, 24, 8/3], [3, 32, 32/27], [4, 40, 40/81]], 1432/81, 否, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)

依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。

本題覆核與來源綁定

覆核日期：2026-10-26

原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。