A224899 - OEIS

%I #29 Jun 02 2022 04:36:35

%S 1,1,8,163,6272,389581,35560448,4479975823,744707981312,

%T 157897753198201,41585725184933888,13318468253704790683,

%U 5097100004294081380352,2297277197389011910783621,1204339195916670860817072128,726625952070893090583192860743

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

%C 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

%H Vaclav Kotesovec, <a href="/A224899/b224899.txt">Table of n, a(n) for n = 0..225</a>

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

%F 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

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

%e where

%e A(x) = 1 + sinh(x) + sinh(2*x)^2 + sinh(3*x)^3 + sinh(4*x)^4 +...

%t 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 *)

%t 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 *)

%o (PARI) {a(n)=n!*polcoeff(sum(k=0, n, sinh(k*x+x*O(x^n))^k), n)}

%o for(n=0, 20, print1(a(n), ", "))

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

%K nonn,nice

%O 0,3

%A _Paul D. Hanna_, Jul 24 2013