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 A361567 #14 Mar 25 2023 08:35:47
%S A361567 1,0,1,6,15,60,555,3150,17745,158760,1399545,10914750,102920895,
%T A361567 1104323220,11249313075,119330961750,1426411411425,17429852840400,
%U A361567 213417453474225,2791671804271350,38524272522310575,537569719902715500,7732658753799054075
%N A361567 Expansion of e.g.f. exp(x^2/2 * (1+x)^2).
%F A361567 a(n) = n! * Sum_{k=0..floor(n/2)} binomial(2*k,n-2*k)/(2^k * k!).
%F A361567 a(0) = 1; a(n) = ((n-1)!/2) * Sum_{k=2..n} k * binomial(2,k-2) * a(n-k)/(n-k)!.
%F A361567 From _Vaclav Kotesovec_, Mar 25 2023: (Start)
%F A361567 a(n) = (n-1)*a(n-2) + 3*(n-2)*(n-1)*a(n-3) + 2*(n-3)*(n-2)*(n-1)*a(n-4).
%F A361567 a(n) ~ 2^(n/4 - 1) * exp(1/128 - 3*2^(-29/4)*n^(1/4) - sqrt(n/2)/16 + 2^(-3/4)*n^(3/4) - 3*n/4) * n^(3*n/4). (End)
%t A361567 Table[n! * Sum[Binomial[2*k,n-2*k]/(2^k * k!), {k,0,n/2}], {n,0,20}] (* _Vaclav Kotesovec_, Mar 25 2023 *)
%o A361567 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x^2/2*(1+x)^2)))
%o A361567 (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!/2*sum(j=2, i, j*binomial(2, j-2)*v[i-j+1]/(i-j)!)); v;
%Y A361567 Cf. A047974, A361568, A361569.
%Y A361567 Cf. A335344, A361278.
%K A361567 nonn
%O A361567 0,4
%A A361567 _Seiichi Manyama_, Mar 16 2023