bibkey: tebni2023a341724
authors: M. Tebni
year: 2023
title: OEIS A341724, Column EGF and prime and dyadic row-divisibility conjectures
doi: null
url: https://oeis.org/A341724
claim: "E.g.f. of column k: x^k / ((1-2*sinh(-x))*k!). For 0 < k < p and p prime, T(p,k) == 0 (mod p). For 0 <= k < n and n = 2^m (m natural number), T(n,k) == 0 (mod n)."
strata_touched:
  - D5/S1/Recurrence/Residue/TebniA341724RowDivisibility
license: citation-only
triage: anchor

OEIS A341724

Verified locator

Source: https://oeis.org/A341724

The entry is unpaginated. Revision 26 attributes both conjectures and the column EGF to Mélika Tebni in its 2023 Comments and Formula blocks:

For 0 < k < p and p prime, T(p,k) == 0 (mod p).

For 0 <= k < n and n = 2^m (m natural number), T(n,k) == 0 (mod n).

E.g.f. of column k: x^k / ((1-2*sinh(-x))*k!).

The array appears in Table 3 of Shannon and Ollerton’s 2021 paper, “A note on Ledin’s summation problem.” That paper supplies the array’s context; the conjectures quoted above are Tebni’s 2023 OEIS contributions. The factorial-normalized coefficient is zero for n<k and one for n=k. The denominator is 1+2 sinh(x), so the zero column begins 1, -2, 8, -50.

The source definitions are literature-attested. The uniform proof of both conjectures is repo-derived: multiply a column by its denominator, use the binomial product formula, prove integrality by induction, and apply prime interior-binomial divisibility or odd-index dyadic-binomial divisibility. The proof includes the prime two, exponent zero and the entire dyadic zero column. It does not assert a valuation formula or a congruence for every composite row.

The bounded literature screen recorded in issue #23158 inspected this OEIS entry, Shannon–Ollerton, Dresden’s arXiv:2206.00115v2 and related Parisse material, alongside repository and formal-conjecture searches. It found no matching settlement in that scope. The literature status is not-found-in-searched-scope; completeness and global priority are unverified.