bibkey: hanna2023a361552
authors: Paul D. Hanna
year: 2023
title: "OEIS A361552: implicit pentagonal unit series and partition parity"
doi: null
url: https://oeis.org/A361552
claim: "Conjecture: a(n)/2 == A000041(n-1) (mod 2) for n >= 1; that is, a(n) is even, and a(n)/2 has the same parity as the number of partitions of n-1, for n >= 1."
strata_touched:
  - D5/S1/Recurrence/Bilateral/HannaPentagonalPartitionParity
license: citation-only
triage: anchor

Hanna’s A361552 partition-parity conjecture

Paul D. Hanna’s OEIS entry, authored March 19, 2023, has revision #14, January 18, 2024. The literal selected comment is:

Conjecture: a(n)/2 == A000041(n-1) (mod 2) for n >= 1; that is, a(n) is even, and a(n)/2 has the same parity as the number of partitions of n-1, for n >= 1.

Its original defining formula is:

G.f. A(x) = Sum_{n>=0} a(n)x^n satisfies the following. (1) 2x = Sum_{n=-oo..+oo} x^(n*(3n+1)/2) * (A(x)^(3n) - 1/A(x)^(3*n+1)).

The immutable locator is https://raw.githubusercontent.com/oeis/oeisdata/5e58e93f885c5f1db9d0523b42a74cc616b16d49/seq/A361/A361552.seq. Its full 3578-byte record has SHA-256 9b7e5254088576b7b061cd7bb443028db7c5a1f74d95a4a22ffdc06dd90c8725. The separate A359914 comparison is outside the selected assertion.

Verified locator

The primary entry is https://oeis.org/A361552. The immutable full source is https://raw.githubusercontent.com/oeis/oeisdata/5e58e93f885c5f1db9d0523b42a74cc616b16d49/seq/A361/A361552.seq.

Classical suppliers

Euler’s pentagonal number theorem is used directly through PowerSeries.WithPiTopology.hasSum_pentagonalSeries and PowerSeries.WithPiTopology.hasProd_one_sub_X_pow. The integer exponent correspondence is two_mul_natCast_pentagonal_neg and natCast_pentagonal. These are the pinned Mathlib proofs in Mathlib.Combinatorics.Enumerative.Pentagonal.Basic and .PowerSeries.

Ordinary partitions are Nat.Partition n, not restricted distinct or odd partitions. The product supplier is Nat.Partition.hasProd_powerSeriesMk_card_restricted at predicate True, from Mathlib.Combinatorics.Enumerative.Partition.Glaisher. The predicate specialization is exactly the full finite set of ordinary partitions. The geometric-factor inverse is the direct application of PowerSeries.WithPiTopology.one_sub_mul_tsum_pow_of_constantCoeff_eq_zero. The unique empty partition is supplied by Nat.Partition.UniquePartitionZero.

Mathlib’s cited supplier files are authored by Weiyi Wang and released under Apache 2.0. This source note retains attribution and short OEIS quotations; it does not redistribute third-party source code. The proof applies the classical identities without a claim of a new Euler or partition method.

Bounded status

The qualified source input checked OEIS A361552 and its direct linked records, exact-ID and related phrase queries on arXiv, Math StackExchange/MathOverflow and GitHub, the extracted Plouffe arXiv:2509.15609v1 paper, and arXiv:2608.11941v2 metadata. It found no exact resolution in that bounded scope. The immutable source is directly accessible and still prints the conjecture. This does not establish absence of private, unindexed or later results, nor first-publication priority.