第 24640 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":[15,30,45,60,75],"x":[1,4],"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":[15,30,45,60,75],"x":[1,4],"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 = [[15, 30, 45, 60, 75]]

derived = [[15, 30, 45, 60, 75], a_n=0 for EVERY n beyond this finite list]

(1/4)^0

OGF ORIGINAL x power0 = 1

OGF term0 coefficient times power := mul(15, 1)

OGF term0 coefficient times power = 15

OGF partial sum through0 := sum(0, 15)

OGF partial sum through0 = 15

(1/4)^1

OGF ORIGINAL x power1 = 1/4

OGF term1 coefficient times power := mul(30, 1/4)

OGF term1 coefficient times power = 15/2

OGF partial sum through1 := sum(15, 15/2)

OGF partial sum through1 = 45/2

(1/4)^2

OGF ORIGINAL x power2 = 1/16

OGF term2 coefficient times power := mul(45, 1/16)

OGF term2 coefficient times power = 45/16

OGF partial sum through2 := sum(45/2, 45/16)

OGF partial sum through2 = 405/16

(1/4)^3

OGF ORIGINAL x power3 = 1/64

OGF term3 coefficient times power := mul(60, 1/64)

OGF term3 coefficient times power = 15/16

OGF partial sum through3 := sum(405/16, 15/16)

OGF partial sum through3 = 105/4

(1/4)^4

OGF ORIGINAL x power4 = 1/256

OGF term4 coefficient times power := mul(75, 1/256)

OGF term4 coefficient times power = 75/256

OGF partial sum through4 := sum(105/4, 75/256)

OGF partial sum through4 = 6795/256

source roles = [[15, 30, 45, 60, 75], 1/4, 4]

derived = [6795/256, 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

([15, 30, 45, 60, 75], 1/4, 4, [[0, 15, 15], [1, 30, 15/2], [2, 45, 45/16], [3, 60, 15/16], [4, 75, 75/256]], 6795/256, 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。