bibkey: hanna2012a219431
authors: Paul D. Hanna
year: 2012
title: "OEIS A219431: logarithmic derivative of square-indexed overpartitions"
doi: null
url: https://oeis.org/A219431
claim: "Conjecture: a(2*n+1) == 2 (mod 4), a(2*n) == 0 (mod 4)."
strata_touched:
  - D5/S3/Combinatorics/Partitions/OverpartitionSquareLogDerivativeModFour
license: citation-only
triage: anchor

Hanna’s conjecture for OEIS A219431

Verified locator

Source: https://oeis.org/A219431

Related generating function: https://oeis.org/A219430

Source statements

OEIS A219431, unpaginated Name line: “Logarithmic derivative of the g.f. of the overpartitions of n^2 (A219430).”

OEIS A219431, unpaginated Comment line: “Conjecture: a(2n+1) == 2 (mod 4), a(2n) == 0 (mod 4).”

OEIS A219430, unpaginated Name line: “Number of overpartitions of n^2; a(n) = A015128(n^2).”

The offset of A219431 is one. Its logarithmic derivative is the series x G'(x) / G(x), where G(x) = sum_n pbar(n^2) x^n. The overpartition coefficient pbar(m) is defined by the product of (1+x^k)/(1-x^k); only factors with 1 <= k <= m contribute to its coefficient at degree m. The formal statement includes integrality of every positive-degree coefficient.

Scope

The modular proof first reduces the overpartition coefficient to twice the number of divisors modulo four. Positive squares have an odd divisor count. The resulting square-sampled series is 1 + 2 sum_{n>=1} x^n modulo four; its logarithmic derivative has coefficient 2n modulo four. The integer unit inverse supplies integrality before reduction.

The completeness of the literature screen and the bibliographic attribution of the standard overpartition congruence remain ASSUMED-UNVERIFIED. Congruences modulo eight and other square-valued sampling sequences are separate questions.