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 A361597 #11 Mar 17 2023 08:29:23
%S A361597 1,1,2,7,40,320,3130,34930,432320,5866840,86816800,1395455600,
%T A361597 24270908200,454897042600,9146979842000,196443726879400,
%U A361597 4486709145318400,108548344109004800,2771885136281060800,74475606190225240000,2099591224223100608000
%N A361597 Expansion of e.g.f. exp( x^3/(6 * (1-x)^3) ) / (1-x).
%F A361597 a(n) = n! * Sum_{k=0..floor(n/3)} binomial(n,3*k)/(6^k * k!).
%F A361597 From _Vaclav Kotesovec_, Mar 17 2023: (Start)
%F A361597 Recurrence: 2*a(n) = 2*(4*n - 3)*a(n-1) - 6*(n-1)*(2*n - 3)*a(n-2) + (n-2)*(n-1)*(8*n - 17)*a(n-3) - 2*(n-3)^2*(n-2)*(n-1)*a(n-4).
%F A361597 a(n) ~ 2^(-7/8) * exp(-1/24 + 5*2^(-15/4)*n^(1/4)/3 - sqrt(n/2)/2 + 2^(7/4)*n^(3/4)/3 - n) * n^(n + 1/8) * (1 + (2637/10240)*2^(3/4)/n^(1/4)). (End)
%t A361597 Table[n! * Sum[Binomial[n,3*k]/(6^k * k!), {k,0,n/3}], {n,0,20}] (* _Vaclav Kotesovec_, Mar 17 2023 *)
%o A361597 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x^3/(6*(1-x)^3))/(1-x)))
%Y A361597 Cf. A002720, A361596.
%Y A361597 Cf. A361573.
%K A361597 nonn
%O A361597 0,3
%A A361597 _Seiichi Manyama_, Mar 16 2023