A binomial coefficient summation

formal-conjectures · oeis · ams-11

A binomial coefficient summation: $a(n) = S(3, n) / S(1, n)$, where for a positive integer $r$
we define
$$S(r,n) = \sum_{k=0}^{\lfloor n/2 \rfloor} \left( \binom{n}{k} - \binom{n}{k-1} \right)^r$$
with $\binom{n}{-1} = 0$.

Let $b(n) = a(2n-1)$. Then the supercongruence $b(n p^k) \equiv b(n p^{k-1}) \pmod{p^{3k}}$
holds for positive integers $n$ and $k$ and all primes $p \ge 5$.
- Zhi-Wei Sun, Nov 16 2019

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/A003162
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/OEIS/3162.lean
Local target: Corpus.OEIS3162.conjecture
Source SHA-256: 2b8f244d72e7da74ed09ee84080ed6045524bae1892f39217d0b6d847c5678fc
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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