ordinary generating function

普通生成函数

Combinatorics · Undergraduate · Mathematics Resources editorial compilation (AI-assisted)

Other names / variants: OGF

Translation evidence: Literature usage

Meaning in context

For a sequence (a_n) indexed by nonnegative integers in a commutative ring R, its ordinary generating function is the formal series ∑ a_n x^n in R[[x]].

交换环 R 中从零开始编号的数列 (a_n)，其普通生成函数是 R[[x]] 中的形式级数 ∑a_n x^n。

Example

The sequence $a_n=3^n$ has OGF $1/(1-3x)$.

Usage and distinctions

Recover a_n as [x^n]A(x); exponential generating functions instead divide each coefficient by n!.

Read the encyclopedia explanation

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Keller and Trotter — Applied Combinatorics (2016)Section 8.1, opening definition and Proposition 8.3 · Checked 2026-10-14

Translation sources

1. 上海交通大学 — CS0901 组合数学（2026）§2.1，定义2.1「普通生成函数」 · Checked 2026-10-14

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