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 A361533 #24 Jun 17 2024 08:49:52
%S A361533 1,0,0,1,4,20,130,980,8400,80920,865200,10164000,130114600,1802600800,
%T A361533 26867640800,428661633400,7288513232000,131558835408000,
%U A361533 2512282795422400,50600743739145600,1071998968264224000,23829055696093648000,554524256514356128000
%N A361533 Expansion of e.g.f. exp(x^3/(6 * (1-x))).
%F A361533 a(n) = 2*(n-1) * a(n-1) - (n-1)*(n-2) * a(n-2) + binomial(n-1,2) * a(n-3) - 2*binomial(n-1,3) * a(n-4) for n > 3.
%F A361533 a(n) ~ 2^(-3/4) * 3^(-1/4) * exp(-5/12 + sqrt(2*n/3) - n) * n^(n - 1/4). - _Vaclav Kotesovec_, Mar 29 2023
%F A361533 From _Seiichi Manyama_, Jun 17 2024: (Start)
%F A361533 a(n) = n! * Sum_{k=0..floor(n/3)} binomial(n-2*k-1,n-3*k)/(6^k * k!).
%F A361533 a(0) = 1; a(n) = ((n-1)!/6) * Sum_{k=3..n} k * a(n-k)/(n-k)!. (End)
%o A361533 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x^3/(6*(1-x)))))
%Y A361533 Cf. A185369, A361545, A361547.
%Y A361533 Cf. A293049.
%K A361533 nonn
%O A361533 0,5
%A A361533 _Seiichi Manyama_, Mar 15 2023