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A229413 Number of set partitions of {1,...,3n} into sets of size at most n.

%I A229413 #9 Dec 10 2020 17:38:35
%S A229413 1,1,76,12644,3305017,1245131903,654277037674,467728049807348,
%T A229413 443694809361207824,544852927413901502514,846359710104516310431744,
%U A229413 1629392161877794034658847500,3819592516111353522143561652540,10738740219595085951726635839975852
%N A229413 Number of set partitions of {1,...,3n} into sets of size at most n.
%H A229413 Alois P. Heinz, <a href="/A229413/b229413.txt">Table of n, a(n) for n = 0..150</a>
%F A229413 a(n) = (3n)! * [x^(3n)] exp(Sum_{j=1..n} x^j/j!).
%F A229413 a(n) = A229223(3n,n).
%p A229413 G:= proc(n, k) option remember; local j; if k>n then G(n, n)
%p A229413       elif n=0 then 1 elif k<1 then 0 else G(n-k, k);
%p A229413       for j from k-1 to 1 by -1 do %*(n-j)/j +G(n-j, k) od; % fi
%p A229413     end:
%p A229413 a:= n-> G(3*n, n):
%p A229413 seq(a(n), n=0..20);
%t A229413 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 A229413 a[n_] := G[3n, n];
%t A229413 a /@ Range[0, 20] (* _Jean-François Alcover_, Dec 10 2020, after _Alois P. Heinz_ *)
%Y A229413 Column k=3 of A229243.
%Y A229413 Cf. A229223.
%K A229413 nonn
%O A229413 0,3
%A A229413 _Alois P. Heinz_, Sep 22 2013