Alphabeta MathHow 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.
FALSE: the Catalan numbers satisfy a constant-coefficient linear recurrence
Statement
False claim: the sequence of Catalan numbers satisfies a linear recurrence with constant coefficients.
Facts & Assumptions
Given: the Catalan numbers and their generating function.
A sequence over a field satisfies an eventual constant-coefficient linear recurrence exactly when its generating function is rational (A coefficient sequence is eventually linearly recurrent if and only if its formal generating function is rational).
The Catalan generating function is not rational ( is not a rational formal power series, so satisfies no eventual constant-coefficient linear recurrence).
The Catalan numbers satisfy with (, with ).
Refutation
If the false claim were true, [L1] would make the Catalan generating function rational.
That contradicts [L2].
The recurrence of [L3] does not rescue the false claim: it is a convolution recurrence, so the next term depends on products of earlier terms rather than on a fixed linear combination.
Remarks
- The tempting mistake is to see the word "recurrence" and forget to ask which kind. The Catalan sequence does have a recurrence, but not the rational-series kind.
Depends on
- $C(x)$ is not a rational formal power series, so $(C_n)$ satisfies no eventual constant-coefficient linear recurrence
- A coefficient sequence is eventually linearly recurrent if and only if its formal generating function is rational
- $C_{n+1}=\sum_{i=0}^{n}C_i\,C_{n-i}$, with $C_0=1$
- The Catalan generating function $C(x)=\sum_{n\ge0}C_nx^n$ in $\mathbb{Q}\llbracket x\rrbracket$
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
27 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
- A. Postnikov (notes by A. Lin), MIT 18.212 Algebraic Combinatorics, Spring 2019 (standard reference, not scraped)