cp's OEIS Frontend

A235134 Expansion of e.g.f. 1/(1 - sinh(2*x))^(1/2).

%I A235134 #25 Jun 24 2025 04:10:47
%S A235134 1,1,3,19,153,1561,19563,289339,4932273,95258161,2055639123,
%T A235134 49019157859,1280056939593,36329281202761,1113449691889083,
%U A235134 36651273215389579,1289577677407798113,48299079453732363361,1918528841276621473443,80559757274836073592499
%N A235134 Expansion of e.g.f. 1/(1 - sinh(2*x))^(1/2).
%C A235134 Generally, for e.g.f. 1/(1-sinh(p*x))^(1/p) we have a(n) ~ n! * p^n / (Gamma(1/p) * 2^(1/(2*p)) * n^(1-1/p) * (arcsinh(1))^(n+1/p)).
%H A235134 G. C. Greubel, <a href="/A235134/b235134.txt">Table of n, a(n) for n = 0..395</a>
%F A235134 a(n) ~ n! * 2^(n-1/4) / (sqrt(Pi*n) * (log(1+sqrt(2)))^(n+1/2)).
%F A235134 a(n) = Sum_{k=0..n} A001147(k) * 2^(n-k) * A136630(n,k). - _Seiichi Manyama_, Jun 24 2025
%t A235134 CoefficientList[Series[1/(1-Sinh[2*x])^(1/2), {x, 0, 20}], x] * Range[0, 20]!
%o A235134 (PARI) x='x+O('x^50); Vec(serlaplace(1/(sqrt(1-sinh(2*x))))) \\ _G. C. Greubel_, Apr 05 2017
%o A235134 (PARI) a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
%o A235134 a001147(n) = prod(k=0, n-1, 2*k+1);
%o A235134 a(n) = sum(k=0, n, a001147(k)*2^(n-k)*a136630(n, k)); \\ _Seiichi Manyama_, Jun 24 2025
%Y A235134 Cf. A006154, A235135.
%Y A235134 Cf. A001147, A001586, A136630, A235131, A380155.
%K A235134 nonn,easy
%O A235134 0,3
%A A235134 _Vaclav Kotesovec_, Jan 03 2014