Central trinomial coefficients

formal-conjectures · oeis · ams-11

Central trinomial coefficients: largest coefficient of $(1 + x + x^2)^n$, which is the coefficient
of $x^n$ in the expansion of $(1 + x + x^2)^n$.

An integer $n > 3$ is prime if and only if $a(n) \equiv 1 \pmod{n^2}$.
We have verified this for $n$ up to $8 \cdot 10^5$, and proved that $a(p) \equiv 1 \pmod{p^2}$
for any prime $p > 3$ (cf. A277640).
- Zhi-Wei Sun, Nov 30 2016

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/A002426
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/OEIS/2426.lean
Local target: Corpus.OEIS2426.conjecture
Source SHA-256: c909f7da52cccc1b31ce3a021dfd85e8928b2312299f217d771f0bc018c3e8ad
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