bibkey: hanna2013a227051
authors: Paul D. Hanna
year: 2013
title: "OEIS A227051: mixed trigonometric and hyperbolic reciprocal"
doi: null
url: https://oeis.org/A227051
claim: "NAME: E.g.f.: 1/(cos(x) - sin(x)*sinh(x)), omitting the zero-valued coefficients of odd powers of x. FORMULA: a(n) == 3 (mod 10) for n>0 (conjecture)."
strata_touched:
  - D5/S1/Recurrence/Residue/HannaMixedTrigHyperbolicModTen
license: citation-only
triage: anchor

Hanna’s A227051 congruence

Verified locator

DOI: null

Source: https://oeis.org/A227051

The source is OEIS A227051, revision 10, with offset 0,2. It has no arXiv identifier or numbered pages. The following are its NAME, first FORMULA, EXAMPLE and PARI coefficient-extraction program, verbatim:

E.g.f.: 1/(cos(x) - sin(x)*sinh(x)), omitting the zero-valued coefficients of odd powers of x.

a(n) == 3 (mod 10) for n>0 (conjecture).

E.g.f.: A(x) = 1 + 3x^2/2! + 53x^4/4! + 2333x^6/6! + 191673x^8/8! + 25307913*x^10/10! +…

(PARI) {a(n)=local(X=x+xO(x^(2n))); (2*n)!*polcoeff(1/(cos(X) - sin(X)sinh(X)), 2n)}

All series in the formal statement are rational formal Taylor series. The index n selects degree 2n and multiplies its coefficient by (2n)!. The statement includes integrality by requiring an integer z equal to the rational coefficient, followed by Int.ModEq 10 z 3. The zeroth coefficient is outside the conjecture. The source’s independent asymptotic formula is outside this congruence assertion.

The bounded literature screen in issue #21582 contains no located settlement. Literature completeness is ASSUMED-UNVERIFIED; two relevant Carlitz full texts have not been checked. The OEIS source sentence is a conjecture; its proof is derived in this repository rather than attributed to the OEIS entry.