bibkey: hanna2013a226839
authors: Paul D. Hanna
year: 2013
title: "OEIS A226839: triangular-exponent exponential generating function"
doi: null
url: https://oeis.org/A226839
claim: "E.g.f.: exp( Sum_{n>=1} x^(n*(n+1)/2) / n! ). a(n) == 1 (mod 3) (conjecture - valid up to n=1024; if true for n>=0, why?)."
strata_touched:
  - D5/S1/Recurrence/Residue/HannaA226839Congruence
license: citation-only
triage: anchor

Hanna’s triangular-exponent EGF

Verified locator

The source is OEIS A226839, returned revision 11, attributed to Paul D. Hanna, June 19, 2013, with offset 0,4. There is no DOI, arXiv identifier, journal version or numbered page asserted for this entry.

NAME, verbatim:

E.g.f.: exp( Sum_{n>=1} x^(n*(n+1)/2) / n! ).

FORMULA, verbatim:

a(n) == 1 (mod 3) (conjecture - valid up to n=1024; if true for n>=0, why?).

The formal interpretation uses rational formal power series: $H(X)={\sum }_{m\ge 1}{X}^{m(m+1)/2}/m!$ and $A(X)=\exp (H(X))$. The source’s factorial coefficient expansion and PARI extraction program identify $a(n)=n![X^n]A$. The natural domain includes zero. The integer witness in the repository result proves integrality together with the congruence; neither is assumed from the source’s integer data.

The retrieved current entry contains no proof of the assertion. The source reader checked the complete current record and the first history page, revisions 11 through 2. A separate prior-work reader checked both history index pages and visible diffs, including revision 1, without reading every standalone revision. Revision 10 changes the lower-index question from n>=1 to n>=0; revision 11 is editorial approval. The reported check through 1024 is the author’s report and was not independently reproduced here. A226838 is a source-linked possible successor, with no qualification or result claimed here.

Prior work and reuse

The bounded source and prior-work screens found no exact settlement: arXiv exact identifier metadata, Crossref, MathOverflow, Math StackExchange, MathDB and indexed Lean searches. The Hanna author/EGF arXiv query located Heninger, Rains and Sloane’s math/0509316v4, whose abstract concerns nth roots of generating functions, a different assertion; its full text was not checked. General-web requests were challenged or irrelevant and supply no absence evidence. Identifier searches do not exclude proofs under another description, unindexed work or unpublished ownership.

Repository and pinned-library searches found the general machinery in IntegralEGFComposition, with the identity-composition lemma owned by QuarticEGFModFour. The repository proof directly reuses these originals. No new general EGF engine or independent escape witness is claimed. The new result’s provenance is FromRepo; OEIS supplies the question and definition, not the repository proof.

The immutable preregistration is issue 23823, original comment. Literature completeness and first-proof priority remain ASSUMED-UNVERIFIED. The bounded screen supports only that no settlement was found in its stated scope.

Escape audit boundary

The typed Reg family preserves the complete coefficient statement and its kernel-checked realization, variation, sensitivity and observational-dependence proofs. Its four-slot binding audit remains unfinished under #23839: declared_unresolved, source.residual_requires_open, with no validated certificate. No registration completion is claimed.