A224899 E.g.f.: Sum_{n>=0} sinh(n*x)^n.

1, 1, 8, 163, 6272, 389581, 35560448, 4479975823, 744707981312, 157897753198201, 41585725184933888, 13318468253704790683, 5097100004294081380352, 2297277197389011910783621, 1204339195916670860817072128, 726625952070893090583192860743

Offset

0,3

Comments

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, 1, 1, 2, 0, 3, 0, ...], with an apparent period of 6. Cf. A245322. - Peter Bala, May 29 2022

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..225

Formula

E.g.f.: Sum_{n>=0} exp(-n^2*x) * (exp(2*n*x) - 1)^n / 2^n.

a(n) ~ sqrt(Pi) * 2^(2*n+1) * n^(2*n+1/2) / (sqrt(3-2*log(2)) * 3^(n+1/2) * exp(2*n) * (log(2))^(2*n+1)). - Vaclav Kotesovec, Oct 28 2014

Example

E.g.f.: A(x) = 1 + x + 8*x^2/2! + 163*x^3/3! + 6272*x^4/4! +...
where
A(x) = 1 + sinh(x) + sinh(2*x)^2 + sinh(3*x)^3 + sinh(4*x)^4 +...

Mathematica

Flatten[{1, Table[Sum[Sum[Binomial[k, j] * (-1)^j * k^n*(k-2*j)^n / 2^k, {j, 0, k}], {k, 0, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, Oct 29 2014 *)
Join[{1}, Rest[With[{nn=20}, CoefficientList[Series[Sum[Sinh[n*x]^n, {n, nn}], {x, 0, nn}], x] Range[0, nn]!]]] (* Harvey P. Dale, May 18 2018 *)

Prog

(PARI) {a(n)=n!*polcoeff(sum(k=0, n, sinh(k*x+x*O(x^n))^k), n)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A122399, A249489, A245322, A220181, A221077, A221078, A198513, A220181, A249459, A195415, A245322, A338040.

Sequence in context: A075387 A143857 A025606 * A281830 A299075 A299840

Adjacent sequences: A224896 A224897 A224898 * A224900 A224901 A224902

Keyword

nonn,nice

Author

Paul D. Hanna, Jul 24 2013

Status

approved