bibkey: bala2022a224899 authors: Peter Bala year: 2022 title: “OEIS A224899, e.g.f. Sum_{n>=0} sinh(nx)^n” doi: null url: https://oeis.org/A224899 claim: “E.g.f.: Sum_{n>=0} sinh(nx)^n. Conjecture: Let p be prime. The sequence obtained by reducing a(n) modulo p for n >= 1 is purely periodic with period p - 1. For example, modulo 7 the sequence becomes [1, 1, 2, 0, 3, 0, 1, 1, 2, 0, 3, 0, …], with an apparent period of 6. Cf. A245322. - Peter Bala, May 29 2022” strata_touched:

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OEIS A224899

Paul D. Hanna authored the entry on April 20, 2013. Peter Bala’s comment of May 29, 2022 states the prime-period conjecture quoted above. The entry has offset 0,3 and begins 1, 1, 8, 163, 6272, 389581, 35560448, 4479975823, 744707981312. The conjecture remains labelled as such in the supplied entry; no proof was located, which does not establish exhaustive absence of a proof.

Expanding sinh(k*x)^k = 2^(-k) (exp(k*x)-exp(-k*x))^k gives the finite coefficient expression H(n,k) = (Sum_{j=0..k} (-1)^(k-j) C(k,j) (k*(2*j-k))^n) / 2^k. The formal sequence is a(n) = Sum_{k=0..n} H(n,k). The e.g.f. identity itself is not formalized. The module proves exact integer division through the Stirling bridge H(n,k) = k^n k! Sum_{r=0..n} C(n,r) (-k)^(n-r) 2^(r-k) S(r,k). Exponent subtraction is natural subtraction; terms with r < k vanish.

The bridge proves factorial divisibility and vanishing for k > n, so only k < p contribute modulo a prime. For odd primes, inverting 2^k yields a fixed sum of positive powers and proves the asserted period p-1, without claiming minimality. At p=2, period one is false: a(1)=1 and a(2)=8 have different residues. The named claim balaConjectureTwo is the exact period-one assertion refuted by bala_conjecture_two_false.

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• URL: https://oeis.org/A224899