Central binomial sum $a(n) = \sum_{k=0}^n \binom{n}{k}^2 \binom{n-k}{k}^2 (-16)^k$

formal-conjectures · oeis · ams-11

The sequence is defined by
$$a(n) = \sum_{k=0}^n \binom{n}{k}^2 \binom{n-k}{k}^2 (-16)^k.$$

If $p$ is a prime with $(p/7) = 1$ and $p = x^2 + 7y^2$ with $x, y$ integers, then
$\sum_{k=0}^{p-1} (-1)^k a(k) \equiv 4x^2 - 2p \pmod{p^2}$.
- _Zhi-Wei Sun_, Jul 17 2010

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/A179537
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://arxiv.org/abs/0911.5665
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/OEIS/179537.lean
Local target: Corpus.OEIS179537.conjecture1
Source SHA-256: 62581f8eee749c200947fde4b3bb4d4823695a1931163d3016320116cb1ed954
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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