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%I #25 Jun 02 2022 04:55:37
%S 1,2,10,92,1351,28982,855100,33214232,1642999501,100843185962,
%T 7520379392890,669760178257172,70211429619908851,8558006664633638942,
%U 1200128210993564085880,191861070874818576596912,34685967730611200643509401,7041037426518318365605795922
%N E.g.f.: Sum_{n>=1} tanh(n*x)^n = Sum_{n>=1} a(n)*4^(n-1)*x^n/n!.
%C Conjecture: Let p be prime. The sequence obtained by reducing a(n) modulo p is purely periodic with period p - 1. For example, modulo 7 the sequence becomes [1, 2, 3, 1, 0, 2, 1, 2, 3, 1, 0, 2, 1, 2, 3, 1, 0, 2, ...], with an apparent period of 6. - _Peter Bala_, May 29 2022
%H Vaclav Kotesovec, <a href="/A195415/b195415.txt">Table of n, a(n) for n = 1..262</a>
%F E.g.f.: Sum_{n>=1} ( 1 - 2/(1+exp(2*n*x)) )^n = Sum_{n>=1} a(n)*4^(n-1)/n!.
%F a(n) ~ c * d^n * n^(2*n + 1/2), where d = 1 / (2 * exp(2) * log(1+sqrt(2))^2) = 0.0871085887239583895519632137900851584739951067757899616766024190... and c = 13.10490857177911562030370300610447966745088413236135355214718... - _Vaclav Kotesovec_, May 31 2022
%F a(n) = A221077(n) / 4^(n-1). - _Vaclav Kotesovec_, Jun 02 2022
%e E.g.f.: A(x) = x + 8*x^2/2! + 160*x^3/3! + 5888*x^4/4! + 345856*x^5/5! +...
%e or, equivalently,
%e A(x) = x + 2*4*x^2/2! + 10*4^2*x^3/3! + 92*4^3*x^4/4! + 1351*4^4*x^5/5! +...
%e where
%e A(x) = tanh(x) + tanh(2*x)^2 + tanh(3*x)^3 + tanh(4*x)^4 + tanh(5*x)^5 +...
%p seq(coeff(n!/4^(n-1)*series(add(tanh(n*x)^n, n = 1..100), x, 101), x, n), n = 1..100); # _Peter Bala_, May 29 2022
%t nmax = 20; Rest[CoefficientList[Series[Sum[Tanh[k*x]^k, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! / 4^(Range[0, nmax] - 1)] (* _Vaclav Kotesovec_, May 31 2022 *)
%o (PARI) {a(n)=local(X=x+x*O(x^n), Egf); Egf=sum(m=1, n, tanh(m*X)^m); n!/4^(n-1)*polcoeff(Egf, n)}
%Y Cf. A122399, A220181, A221077, A221078, A224899, A245322, A338040.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Sep 17 2011
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