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A229229 Number of set partitions of {1,...,n^2} into sets of size at most n.

%I A229229 #15 Feb 15 2017 11:08:10
%S A229229 1,1,10,12644,6631556521,3282701194678476257,
%T A229229 3025262978042089315465899013351,
%U A229229 9292286146024114784457467780130028866860171013,158655194198118596873150397161518177395553186289541468458000908304
%N A229229 Number of set partitions of {1,...,n^2} into sets of size at most n.
%H A229229 Alois P. Heinz, <a href="/A229229/b229229.txt">Table of n, a(n) for n = 0..24</a>
%F A229229 a(n) = (n^2)! * [x^(n^2)] exp(Sum_{j=1..n} x^j/j!).
%F A229229 a(n) = A229223(n^2,n).
%e A229229 a(2) = 10: 1/2/3/4, 12/3/4, 13/2/4, 14/2/3, 1/23/4, 1/24/3, 1/2/34, 12/34, 13/24, 14/23.
%p A229229 G:= proc(n, k) option remember; local j; if k>n then G(n, n)
%p A229229       elif n=0 then 1 elif k<1 then 0 else G(n-k, k);
%p A229229       for j from k-1 to 1 by -1 do %*(n-j)/j +G(n-j, k) od; % fi
%p A229229     end:
%p A229229 a:= n-> G(n^2, n):
%p A229229 seq(a(n), n=0..10);
%t A229229 G[n_, k_] := G[n, k] = Module[{j, pc}, Which[k>n, G[n, n], n==0, 1, k<1, 0, True, pc = G[n-k, k]; For[j = k-1, j >= 1, j--, pc = pc*(n-j)/j + G[n-j, k]]; pc]]; a[n_] := G[n^2, n]; Table[a[n], {n, 0, 10}] (* _Jean-François Alcover_, Feb 15 2017, translated from Maple *)
%Y A229229 Main diagonal of A229243.
%Y A229229 Cf. A229223.
%K A229229 nonn
%O A229229 0,3
%A A229229 _Alois P. Heinz_, Sep 16 2013