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 A361573 #17 May 20 2023 12:26:22
%S A361573 1,0,0,1,12,120,1210,13020,152880,1975960,28148400,440470800,
%T A361573 7525441000,139375236000,2778421245600,59239029249400,
%U A361573 1343609515248000,32274288638592000,818014942318974400,21809788600885084800,610079100418595808000,17863467401461938256000
%N A361573 Expansion of e.g.f. exp(x^3/(6 * (1 - x)^3)).
%F A361573 a(n) = n! * Sum_{k=0..floor(n/3)} binomial(n-1,n-3*k)/(6^k * k!).
%F A361573 a(0) = 1; a(n) = ((n-1)!/6) * Sum_{k=3..n} (-1)^(k-3) * k * binomial(-3,k-3) * a(n-k)/(n-k)!.
%F A361573 a(n) ~ 2^(-9/8) * exp(-1/24 + 5*2^(1/4)*n^(1/4)/48 - sqrt(2*n)/4 + 2^(7/4)*n^(3/4)/3 - n) * n^(n - 1/8) * (1 - 1203*2^(3/4)/(10240*n^(1/4))). - _Vaclav Kotesovec_, Mar 29 2023
%t A361573 With[{nn=30},CoefficientList[Series[Exp[x^3/(6(1-x)^3)],{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, May 20 2023 *)
%o A361573 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x^3/(6*(1-x)^3))))
%o A361573 (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!/6*sum(j=3, i, (-1)^(j-3)*j*binomial(-3, j-3)*v[i-j+1]/(i-j)!)); v;
%Y A361573 Cf. A000262, A335344, A361577.
%K A361573 nonn
%O A361573 0,5
%A A361573 _Seiichi Manyama_, Mar 16 2023