This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A383837 #20 May 17 2025 05:03:30
%S A383837 1,1,16,820,87296,15857205,4390088704,1721255653656,907673633095680,
%T A383837 619593964021650475,531571294549842067456,559896149105493602658256,
%U A383837 710322778732936488128872448,1068386732538408106621063668220,1879866814874817967233600382304256
%N A383837 a(n) = (3*n)!/n! * [x^(3*n)] sinh(x)^n.
%H A383837 Vaclav Kotesovec, <a href="/A383837/b383837.txt">Table of n, a(n) for n = 0..200</a>
%F A383837 a(n) = [x^n] 1/Product_{k=0..floor(n/2)} (1 - (n-2*k)^2*x).
%F A383837 a(n) = (1/(2^n*n!)) * Sum_{k=0..n} (-1)^k * (n-2*k)^(3*n) * binomial(n,k).
%F A383837 a(n) ~ c * d^n * n^(2*n - 1/2), where d = 1.35572032955623014748562257137412853926900571707993382361... and c = 0.81034327454108346293530087910356437429774959841653144433... - _Vaclav Kotesovec_, May 13 2025
%F A383837 In closed form, a(n) ~ r^(r*n) * (1 + 2*r)^(3*n+1) * exp(n) * n^(2*n - 1/2) / (sqrt(Pi*(1 - 8*r - 8*r^2)) * 2^(n - 1/2) * (1+r)^((1+r)*n)), where r = 0.002562299585216598238663221142585901101711497682846... is the positive real root of the equation exp(2*arctanh(1 + 2*r) - 6/(1 + 2*r)) = -1. - _Vaclav Kotesovec_, May 17 2025
%t A383837 Join[{1}, Table[Sum[(-1)^k * (n-2*k)^(3*n) * Binomial[n, k] / (2^n*n!), {k,0,n}],{n,1,20}]] (* _Vaclav Kotesovec_, May 13 2025 *)
%o A383837 (PARI) a(n) = sum(k=0, n, (-1)^k*(n-2*k)^(3*n)*binomial(n, k))/(2^n*n!);
%Y A383837 Main diagonal of A381512.
%Y A383837 Cf. A298851, A381459.
%K A383837 nonn
%O A383837 0,3
%A A383837 _Seiichi Manyama_, May 11 2025