convolution of sequences

卷积

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

Other names / variants: Cauchy product coefficients

Translation evidence: Literature usage

Meaning in context

For sequences (a_n) and (b_n) in a commutative ring indexed by n ≥ 0, their convolution is c_n = ∑{k=0}^n a_k b{n−k}.

从零开始编号的两个交换环值数列，其卷积定义为 c_n=∑{k=0}^n a_k b{n−k}。

Example

Convolving the constant sequences an=bn=1a_n=b_n=1 gives cn=n+1c_n=n+1.

Usage and distinctions

This is discrete ordinary convolution; multiplying EGFs produces binomial-weighted counts instead.

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）§2.2末，普通生成函数常用操作表，「卷积」行 · Checked 2026-10-14
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