bibkey: hanna2022a356508
authors: Paul D. Hanna
year: 2022
title: "OEIS A356508, product-partition generating-function congruence"
doi: null
url: https://oeis.org/A356508
claim: "Conjecture: a(n) == 2 (mod 4) at n = 2*k for all k > 0 that have an odd number of partitions (A052002), otherwise a(n) == 0 (mod 4) when n > 0."
strata_touched:
  - D5/S1/Recurrence/Invariants/HannaProductPartitionModFour
license: citation-only
triage: anchor

OEIS A356508 product-partition congruence

P. D. Hanna’s OEIS A356508 entry was created 11 August 2022. Its Name field states verbatim:

G.f. A(x) satisfies: 2 = Product_{n>=1} (1 + x^n*A(x)) * (1 + x^(n-1)/A(x)).

Its Comments field states verbatim:

Conjecture: a(n) == 2 (mod 4) at n = 2*k for all k > 0 that have an odd number of partitions (A052002), otherwise a(n) == 0 (mod 4) when n > 0.

The entry has offset zero and begins with 1, 4, 14, 84, 444, 2928, 18214, 125428, 844534, 5989816, …. The formalization interprets the product coefficientwise: at degree m, factors with n at least m+2 are one modulo X^(m+1), and A⁻¹ is the formal inverse of the rational series A.

Verified locator

DOI: null

Source: https://oeis.org/A356508

Settlement scope

The proof constructs the unique rational solution by coefficient contraction and shows its coefficients are integers. After mapping to ZMod 4, the candidate 2 * partitionSeries(X^2) - 1 squares to one. Pairing the two factors reduces the product to the reciprocal partition identity, giving coefficient residue 2 times the ordinary partition number at n/2 for even n and zero for odd n.

Residues modulo 8, replacement of the constant 2 by a general scalar, and Jacobi triple-product or asymptotic questions are outside this settlement.

The finite check in the-omega-institute/trureturing-experiments/docs/reports/oeis-a356508-hanna-product-partition-mod4 was run at commit b545d0b3f4226d80274b80c06cf01c1153c12957; its check.py has SHA-256 00c00982ce21ca8c40c9e9c216f48bffc4a79dbf590b1eadecde697c815f0ab3 and python3 check.py 120 b356508.txt exited 0 with RESULT PASS.

Literature completeness is ASSUMED-UNVERIFIED.