第 24636 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":[11,22,33,44,55],"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":[11,22,33,44,55],"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 = [[11, 22, 33, 44, 55]]

derived = [[11, 22, 33, 44, 55], a_n=0 for EVERY n beyond this finite list]

(1/3)^0

OGF ORIGINAL x power0 = 1

OGF term0 coefficient times power := mul(11, 1)

OGF term0 coefficient times power = 11

OGF partial sum through0 := sum(0, 11)

OGF partial sum through0 = 11

(1/3)^1

OGF ORIGINAL x power1 = 1/3

OGF term1 coefficient times power := mul(22, 1/3)

OGF term1 coefficient times power = 22/3

OGF partial sum through1 := sum(11, 22/3)

OGF partial sum through1 = 55/3

(1/3)^2

OGF ORIGINAL x power2 = 1/9

OGF term2 coefficient times power := mul(33, 1/9)

OGF term2 coefficient times power = 11/3

OGF partial sum through2 := sum(55/3, 11/3)

OGF partial sum through2 = 22

(1/3)^3

OGF ORIGINAL x power3 = 1/27

OGF term3 coefficient times power := mul(44, 1/27)

OGF term3 coefficient times power = 44/27

OGF partial sum through3 := sum(22, 44/27)

OGF partial sum through3 = 638/27

(1/3)^4

OGF ORIGINAL x power4 = 1/81

OGF term4 coefficient times power := mul(55, 1/81)

OGF term4 coefficient times power = 55/81

OGF partial sum through4 := sum(638/27, 55/81)

OGF partial sum through4 = 1969/81

source roles = [[11, 22, 33, 44, 55], 1/3, 4]

derived = [1969/81, 是]

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

([11, 22, 33, 44, 55], 1/3, 4, [[0, 11, 11], [1, 22, 22/3], [2, 33, 11/3], [3, 44, 44/27], [4, 55, 55/81]], 1969/81, 是, 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。