TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-26

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Pointing translates to $x A^{'} (x)$

Statement

Let $A$ be a combinatorial class with ordinary generating function

$A (x) = \sum_{n \geq 0} a_{n} x^{n} .$

Then

$OGF (Θ A) = x A^{'} (x) .$

Facts & Assumptions

Given: A combinatorial class $A$ with counting sequence $(a_{n})$ and ordinary generating function $A (x)$ .

[L1]

The formal derivative of $\sum_{n \geq 0} a_{n} x^{n}$ is $\sum_{n \geq 1} n a_{n} x^{n - 1}$ (The formal derivative $D (\sum a_{n} x^{n}) = \sum_{n \geq 1} n a_{n} x^{n - 1}$ ).

Proof

technique · direct

1.1

For each $n \geq 0$ , every size- $n$ object of $A$ contributes exactly $n$ pointed objects of size $n$ , one for each distinguished position. Hence the size- $n$ layer of $Θ A$ has cardinality $n a_{n}$ .

given

2.1

Therefore $OGF (Θ A) = \sum_{n \geq 0} n a_{n} x^{n} = x \sum_{n \geq 1} n a_{n} x^{n - 1} = x A^{'} (x)$ , using [L1] for the last equality.

step 1.1L1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics (standard reference, not scraped)
Stephen Melczer, An Invitation to Enumeration, Chapter 5: Combinatorial Constructions (standard reference, not scraped)

Pointing translates to xA′(x)xA′(x)