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A235135 Expansion of e.g.f. 1/(1 - sinh(3*x))^(1/3).

%I A235135 #19 Jun 24 2025 04:11:42
%S A235135 1,1,4,37,424,6241,113824,2460277,61504384,1746727201,55545439744,
%T A235135 1955176596517,75470959673344,3169939381277761,143927870364811264,
%U A235135 7024566555751464757,366742587098140770304,20394984041632355113921,1203587891190987380752384,75124090160952970927512997
%N A235135 Expansion of e.g.f. 1/(1 - sinh(3*x))^(1/3).
%C A235135 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)).
%F A235135 a(n) ~ n! * 3^n / (Gamma(1/3) * 2^(1/6) * n^(2/3) * (log(1+sqrt(2)))^(n+1/3)).
%F A235135 a(n) = Sum_{k=0..n} A007559(k) * 3^(n-k) * A136630(n,k). - _Seiichi Manyama_, Jun 24 2025
%t A235135 CoefficientList[Series[1/(1-Sinh[3*x])^(1/3), {x, 0, 20}], x] * Range[0, 20]!
%o A235135 (PARI) a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
%o A235135 a007559(n) = prod(k=0, n-1, 3*k+1);
%o A235135 a(n) = sum(k=0, n, a007559(k)*3^(n-k)*a136630(n, k)); \\ _Seiichi Manyama_, Jun 24 2025
%Y A235135 Cf. A006154, A235134.
%Y A235135 Cf. A007559, A007788, A136630, A235132.
%K A235135 nonn,easy
%O A235135 0,3
%A A235135 _Vaclav Kotesovec_, Jan 03 2014