ExampleConstruction: AI-generatedVerification: 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.

The Catalan numbers through $C_{6}$ , from the recurrence and from the closed formula

Example

The Catalan numbers through $C_{6}$ are:

$n$	from the recurrence	$(\binom{2 n}{n})$	from $(n + 1) C_{n} = (\binom{2 n}{n})$
$0$	$1$	$1$	$1$
$1$	$1$	$2$	$1$
$2$	$2$	$6$	$2$
$3$	$5$	$20$	$5$
$4$	$14$	$70$	$14$
$5$	$42$	$252$	$42$
$6$	$132$	$924$	$132$

Facts & Assumptions

Given: the Catalan recurrence and the closed formula.

[L1]

$C_{n + 1} = \sum_{i = 0}^{n} C_{i} C_{n - i}$ with $C_{0} = 1$ ( $C_{n + 1} = \sum_{i = 0}^{n} C_{i} C_{n - i}$ , with $C_{0} = 1$ ).

[L2]

$(n + 1) C_{n} = (\binom{2 n}{n})$ ( $(n + 1) C_{n} = (\binom{2 n}{n})$ ).

Verification

technique · direct

1.1

Starting from $C_{0} = 1$ , the recurrence [L1] gives successively $C_{1} = 1$ , $C_{2} = 2$ , $C_{3} = 5$ , $C_{4} = 14$ , $C_{5} = 42$ and $C_{6} = 132$ .

1.2

The central binomial coefficients in the third column are $(\binom{0}{0}) = 1$ , $(\binom{2}{1}) = 2$ , $(\binom{4}{2}) = 6$ , $(\binom{6}{3}) = 20$ , $(\binom{8}{4}) = 70$ , $(\binom{10}{5}) = 252$ and $(\binom{12}{6}) = 924$ .

given

2.1

Dividing the third column by $n + 1$ as [L2] prescribes gives exactly the second column again, so the two routes agree term by term.

L2step 1.1step 1.2∎

Remarks

The table is the finite check behind the three proofs on the A page: every one of them lands on the same sequence before any general theorem is applied.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

25 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)

The Catalan numbers through C6C6​, from the recurrence and from the closed formula