Apéry numbers

formal-conjectures · oeis · ams-11 · ams-12

Apéry numbers:
$$a(n) = \sum_{k=0}^n \binom{n}{k}^2 \binom{n+k}{k}$$

For each $n = 1, 2, 3, \dots$ the polynomial
$a_n(x) = \sum_{k=0}^n \binom{n}{k}^2 \binom{n+k}{k} x^k$
is irreducible over the field of rational numbers.
- Zhi-Wei Sun, Mar 21 2013

Mathematical status
Open: marked research open in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11).

Formal availability
A Prop definition is supplied for the pinned Lean 4.33.1 environment and must be accepted by the deployed verifier before it is available.

Sources and provenance
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://oeis.org/A005258
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/OEIS/5258.lean
Local target: Corpus.OEIS5258.conjecture
Source SHA-256: e27906cbab339ede156388a225dca386e7212941f8f17169086d257d5ae3b492
Lean v4.33.1; Mathlib 0df444a360eaa60ab8c11dca51a86af692955474; policy kernel-replay-v1.
The deployed accepted environment records the actual immutable verifier image.

Research discussions are not verified proofs. A formal target specifies a precise statement; accepting a target does not prove it. Checked results apply to their exact statements and pinned environments.

Public JSON record