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 A381344 #16 Apr 19 2025 06:56:01
%S A381344 1,1,2,12,72,500,4560,47936,565376,7572240,112838400,1844425792,
%T A381344 32910332928,636463467328,13251265570816,295598326909440,
%U A381344 7034150340034560,177843592245969152,4760839037033054208,134528586280018721792,4001489050575059025920,124973219149863342633984
%N A381344 Expansion of e.g.f. 1/( 1 - x * cosh(sqrt(2)*x) ).
%C A381344 As stated in the comment of A185951, A185951(n,0) = 0^n.
%F A381344 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} 2^k * (2*k+1) * binomial(n,2*k+1) * a(n-2*k-1).
%F A381344 a(n) = Sum_{k=0..n} k! * 2^((n-k)/2) * A185951(n,k).
%F A381344 a(n) ~ sqrt(Pi) * 2^(n/2 + 1) * n^(n + 1/2) / ((cosh(r) + r*sinh(r)) * exp(n) * r^(n+1)), where r = 0.95090803593755778120914299086438615849657408871... is the root of the equation r*cosh(r) = sqrt(2). - _Vaclav Kotesovec_, Apr 19 2025
%o A381344 (PARI) a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
%o A381344 a(n) = sum(k=0, n, k!*2^((n-k)/2)*a185951(n, k));
%Y A381344 Cf. A205571, A352252, A381280, A381281, A381282, A381283, A381345.
%Y A381344 Cf. A185951.
%K A381344 nonn
%O A381344 0,3
%A A381344 _Seiichi Manyama_, Feb 20 2025