bibkey: hanna2024a222014 authors: Paul D. Hanna year: 2024 title: “OEIS A222014, factorial square-exponent product-sum generating function and parity conjecture” doi: null url: https://oeis.org/A222014 claim: “Conjecture: a(n) is odd iff n = 2^k - 1 for some k >= 0.” strata_touched:

• D5/S1/Recurrence/Parity/FactorialSquareExponentSumParity license: citation-only triage: anchor

OEIS A222014

Paul D. Hanna’s comment of December 6, 2024 records the parity conjecture quoted above. The entry defines the ordinary generating series by

A(x) = Sum_{n>=0} n! * x^n * A(x)^(n^2) / Product_{k=1..n} (1 + k*x*A(x)^n).

At coefficient degree N, the formal equation is represented by the exact finite window 0 <= r <= N. Every denominator factor has constant coefficient one, so division is its formal unit inverse. The explicit factor X^r makes all terms with r > N invisible to coefficient N.

The exponent-parametric contraction contains this square-exponent term and the triangular-exponent A222013 term as exact specializations. After reduction modulo two, every term with r >= 2 vanishes because 2 divides r!; the two remaining terms yield C = 1 + X*C^2. The A222013 parity theorem then identifies the odd coefficients with the indices one below a power of two.

The directed formal dependency is D5/S1/Recurrence/Parity/FactorialSquareExponentSumParity to D5/S1/Recurrence/Parity/FactorialProductSumCatalanParity, specifically through mod_two_identity and hanna_conjecture.

Verified locator

• URL: https://oeis.org/A222014