Numerator of a sum involving binomial coefficients

formal-conjectures · oeis · ams-11

$a(n)$ is the numerator of
$\sum_{k = 1}^n \frac{1}{k^3} \binom{n}{k}^2 \binom{n+k}{k}^2$ for $n \ge 1$
with $a(0) = 0$.

We conjecture that $u(p-1) == 0 (mod p^4)$ for all primes $p$,
with a finite number of exceptions that depend on $m$.

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/A357513
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/OEIS/357513.lean
Local target: Corpus.OEIS357513.general_supercongruence
Source SHA-256: 61eef598d8ea2ecf9c31a7963b9b5b7861af8fbf8b903a51884b6ec83dbf89c0
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.

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