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%I #32 Jun 02 2022 08:03:59
%S 1,1,8,160,5888,345856,29677568,3502489600,544181977088,
%T 107675615297536,26435436140822528,7885689342279024640,
%U 2809177794704769548288,1177952320402008693538816,574318105367992485583781888,322156963576521588458420961280,206009256195720974104252003647488
%N E.g.f.: Sum_{n>=0} tanh(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, 6, 1, 0, 4, 1, 1, 6, 1, 0, 4, 1, 1, 6, 1, 0, 4 ...], with an apparent period of 6. - _Peter Bala_, Jun 01 2022
%H Vaclav Kotesovec, <a href="/A221077/b221077.txt">Table of n, a(n) for n = 0..220</a>
%F E.g.f.: Sum_{n>=0} (exp(2*n*x) - 1)^n / (exp(2*n*x) + 1)^n.
%F a(n) ~ c * 2^n * (n!)^2 / (sqrt(n) * (log(1+sqrt(2)))^(2*n)), where c = 0.521427744491499132141002572969819345522922990165233786929882335275903215... - _Vaclav Kotesovec_, Nov 05 2014, updated Jun 02 2022
%e E.g.f.: A(x) = 1 + x + 8*x^2/2! + 160*x^3/3! + 5888*x^4/4! + 345856*x^5/5! +...
%e where
%e A(x) = 1 + tanh(x) + tanh(2*x)^2 + tanh(3*x)^3 + tanh(4*x)^4 + tanh(5*x)^5 +...
%t nmax = 20; CoefficientList[Series[1 + Sum[Tanh[k*x]^k, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* _Vaclav Kotesovec_, May 31 2022 *)
%t Join[{1}, Table[Sum[2^n * k^n * Sum[(-1)^j * Binomial[k, j] * Sum[(-1)^m * Binomial[j + m - 1, m] * StirlingS2[n, m] * m! / 2^m, {m, 1, n}], {j, 0, k}], {k, 0, n}], {n, 1, 20}]] (* _Vaclav Kotesovec_, Jun 01 2022 *)
%o (PARI) {a(n)=local(X=x+x*O(x^n), Egf); Egf=sum(m=0, n, tanh(m*X)^m); n!*polcoeff(Egf, n)}
%o for(n=0,20,print1(a(n),", ") )
%o (PARI) {a(n)=local(X=x+x*O(x^n), Egf); Egf=sum(m=0, n, (exp(2*m*X)-1)^m/(exp(2*m*X)+1)^m); n!*polcoeff(Egf, n)}
%o for(n=0,20,print1(a(n),", ") )
%Y Cf. A122399, A195415, A220181, A221078, A221079, A224899, A249489, A245322, A338040.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Dec 31 2012
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