Metadata
bibkey: hanna2016a277403
authors: Paul D. Hanna
year: 2016
title: "OEIS A277403 — E.g.f. satisfies: A(x - Integral A(x) dx) = x + Integral A(x) dx."
doi: null
url: https://oeis.org/A277403
claim: "a(n) is divisible by 10 for n>2 (conjecture)."
strata_touched:
- D5/S1/Recurrence/Residue/HannaIntegralCompositionDivTen
license: citation-only
triage: anchor
Verified locator
Source: https://oeis.org/A277403
Plain-text record: https://oeis.org/search?fmt=text&q=id:A277403 Revision #21, 20 October 2019 09:38:03; authored 14 October 2016. The source is an OEIS entry without a DOI, source arXiv identifier, or page numbers.
Source statement
The NAME field reads verbatim:
E.g.f. satisfies: A(x - Integral A(x) dx) = x + Integral A(x) dx.
The sole COMMENTS conjecture reads verbatim:
a(n) is divisible by 10 for n>2 (conjecture).
The FORMULA field introduces the integral:
Let G(x) = Integral A(x) dx, then e.g.f. A(x) also satisfies:
The offset is one. The example begins
A(x) = x + 2*x^2/2! + 10*x^3/3! + 90*x^4/4! + ....
It fixes the zero constant term of A and the zero integration constant.
In rational formal series the interpretation is
A(0)=G(0)=0, G'=A, and A.subst (X-G)=X+G.
The sequence coefficient is n! times the ordinary degree-n coefficient.
Integrality is proved along with the congruence rather than assumed.
Mathematical scope
The repository proof constructs a unique integral triangular coefficient solution.
The finite polynomial P=X+X^2, with primitive J=X^2/2+X^3/3, has complete
residual exponential coefficients -10,-10,40,80 in degrees three through six.
Uniqueness after reduction modulo ten proves the all-index congruence.
This proof is not attributed to the OEIS entry.
The preregistration and bounded literature screen are issue #21746. No settlement was found in the specified accessible searches. Bibliographic completeness remains ASSUMED-UNVERIFIED, including unavailable OpenAlex and Semantic Scholar citation searches and exhaustive arXiv coverage.
Independent computation
The experiment is in the-omega-institute/trureturing-experiments, commit
de4bf43c9af7cabb0e9a46c584267cf950fb8beb, directory
docs/reports/hanna-a277403-div10/.
Its check.py has SHA-256
e1f1bb9fb343a3ac3f40d8e878fefc07e49ca10d4f6977701a81e4cff159e777.
The command python3 docs/reports/hanna-a277403-div10/check.py exits zero
with final line ALL_CHECKS_OK, as recorded in #21746.
It checks the source terms, rational substitution through degree 24,
residues through degree 60, and the complete polynomial residual.
These finite checks support the interpretation; the all-index conclusion
uses the kernel-checked induction.