Coefficients of Π_k>0 (1 - x^k/k!)

The coefficients c(n) of A(x)^2 = (Σ_n ≥ 0 a(n) x^n)^2 differ in sign from c(n-1) if and only if n is a triangular number. - _Peter Bala_, Mar 17 2022

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-oeis-185895,
  title        = {Coefficients of Π_k>0 (1 - x^k/k!)},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-185895}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

The coefficients $c (n)$ of $A (x)^{2} = (\sum_{n \geq 0} a (n) x^{n})^{2}$ differ in sign from $c (n - 1)$ if and only if $n$ is a triangular number.

_Peter Bala_, Mar 17 2022

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

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«185895».

theorem conjecture2 (n : ℕ) (hn : 0 < n) :
    c n * c (n - 1) < 0 ↔ IsTriangular n

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

A185895

Source and licence

Imported from Formal Conjectures (OEIS), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.