bibkey: tebni2023a364822
authors: M. Tebni
year: 2023
title: OEIS A364822, Expansion of e.g.f. cosh(x) / (1 - 2*sinh(x))
doi: null
url: https://oeis.org/A364822
claim: "Expansion of e.g.f. cosh(x) / (1 - 2*sinh(x)). Conjectures: For p prime (p > 2), a(p) == 2 (mod p). For n = 2^m (m natural number), a(n) == 1 (mod n)."
strata_touched:
  - D5/S1/Recurrence/Residue/TebniA364822
  - D5/S1/Recurrence/Residue/TebniA364822PowerOfTwo
license: citation-only
triage: anchor

OEIS A364822

Verified locator

Source: https://oeis.org/A364822

The entry is unpaginated. Revision 25 supplies the following sentences.

Name: “Expansion of e.g.f. cosh(x) / (1 - 2*sinh(x)).”

Comments: “Conjectures: For p prime (p > 2), a(p) == 2 (mod p).” “For n = 2^m (m natural number), a(n) == 1 (mod n).”

The index starts at zero, with coefficients 1, 2, 9, 56, 465, 4832, 60249, 876416. The rational formal series for cosh and sinh have coefficients 1/n! at even and odd indices respectively, and zero otherwise. Multiplying the nth ordinary coefficient of their quotient by n! gives a(n).

The two Lean results prove integrality and the stated congruences for every odd prime and every natural exponent respectively. The proof uses the binomial recurrence obtained by multiplying by 1 - 2 sinh(x). Substitution of 2 sinh into the geometric series preserves integral factorial-scaled coefficients; the binomial product formula gives integrality after multiplication by cosh. Interior binomial coefficients vanish at a prime index, and odd-index binomial coefficients vanish modulo a power of two. The case m = 0 has modulus one. No uniform exact valuation theorem for all even indices is asserted.

The experiment is the-omega-institute/trureturing-experiments, commit 43e699d21346a1c6156fbac2901ce602af85ccd9, docs/reports/oeis-a364822-cosh-sinh-egf-congruences/check.py, SHA256 52f51c8b6cb41d6525e4b991d6c081a96f6bd9ff9783c9ab6888857e2919ce24. The recorded command python3 check.py 256 100001 exited 0 with RESULT PASS; it checked both congruences through 256 and the parity count through odd r <= 100001. These are computed readings, independent of the unbounded Lean statements.

The literature screen recorded in issues #18643 and #18644 includes the OEIS entry and Fried’s papers arXiv:2410.07237, 2606.09913 and 2607.24832. No proof of these conjectures was found in that screen. Its completeness is ASSUMED-UNVERIFIED; no priority claim is made.