Coefficients of $\prod_{k>0} (1 - x^k/k!)$

formal-conjectures · oeis · ams-11

The sequence $a(n)$ has exponential generating function
$$E(x) = \prod_{k=1}^\infty \left(1 - \frac{x^k}{k!}\right),$$
so that $a(n) = n! [x^n] \prod_{k=1}^n \left(1 - \frac{x^k}{k!}\right)$.

$a(n)$ differs in sign from $a(n-1)$ if and only if $n$ is a triangular number
(checked up to $n = 1225 = (50 \cdot 51)/2$).
- _Peter Bala_, Mar 17 2022

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/A185895
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/OEIS/185895.lean
Local target: Corpus.OEIS185895.conjecture1
Source SHA-256: 96de652d627db8b041caef6fc041153a915df138ea03366dc4709e7de6ef8ecf
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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