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A229414 Number of set partitions of {1,...,3n} into sets of size at most 3.

%I A229414 #12 Dec 10 2020 17:38:04
%S A229414 1,5,166,12644,1680592,341185496,97620050080,37286121988256,
%T A229414 18280749571449664,11168256342434121152,8306264068494786829696,
%U A229414 7380771881944947770497280,7715405978050522488223499776,9365880670184268387214967727104,13058232187415887547449498864463872
%N A229414 Number of set partitions of {1,...,3n} into sets of size at most 3.
%H A229414 Alois P. Heinz, <a href="/A229414/b229414.txt">Table of n, a(n) for n = 0..200</a>
%F A229414 a(n) = (3n)! * [x^(3n)] exp(x + x^2/2 + x^3/6).
%F A229414 a(n) = A001680(3n) = A229223(3n,3).
%p A229414 a:= proc(n) option remember; `if`(n<3, [1, 5, 166][n+1],
%p A229414       ((108*n^2-72*n+4)*a(n-1)-6*(n-1)*(3*n-5)*(27*n^2-48*n+10)*a(n-2)
%p A229414        +9*(n-1)*(n-2)*(3*n-1)*(3*n-7)*(3*n-5)*(3*n-8)*a(n-3))/8)
%p A229414     end:
%p A229414 seq(a(n), n=0..20);
%t A229414 G[n_, k_] := G[n, k] = Module[{j, g}, Which[k > n, G[n, n], n == 0, 1, k < 1, 0, True, g = G[n - k, k]; For[j = k - 1, j >= 1, j--, g = g(n-j)/j + G[n - j, k]]; g]];
%t A229414 a[n_] := G[3n, 3];
%t A229414 a /@ Range[0, 20] (* _Jean-François Alcover_, Dec 10 2020, after _Alois P. Heinz_ in A229243 *)
%Y A229414 Row n=3 of A229243.
%K A229414 nonn
%O A229414 0,2
%A A229414 _Alois P. Heinz_, Sep 22 2013