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A227223 Number of set partitions of {1,...,2^n} into sets of size at most n.

%I A227223 #11 Dec 30 2013 04:19:31
%S A227223 0,1,10,2780,6631556521,71669271794142235712392433,
%T A227223 78417479379491793666843945562521255790293292270929676484784808001
%N A227223 Number of set partitions of {1,...,2^n} into sets of size at most n.
%H A227223 Alois P. Heinz, <a href="/A227223/b227223.txt">Table of n, a(n) for n = 0..9</a>
%F A227223 a(n) = (2^n)! * [x^(2^n)] exp(Sum_{j=1..n} x^j/j!).
%F A227223 a(n) = A229223(2^n,n).
%e A227223 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 A227223 G:= proc(n, k) option remember; local j; if k>n then G(n, n)
%p A227223       elif n=0 then 1 elif k<1 then 0 else G(n-k, k);
%p A227223       for j from k-1 to 1 by -1 do %*(n-j)/j +G(n-j, k) od; % fi
%p A227223     end:
%p A227223 a:= n-> G(2^n, n):
%p A227223 seq(a(n), n=0..7);
%t A227223 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]]; a[n_] := G[2^n, n]; Table[a[n], {n, 0, 7}] (* _Jean-François Alcover_, Dec 30 2013, translated from Maple *)
%Y A227223 Cf. A229223.
%K A227223 nonn
%O A227223 0,3
%A A227223 _Alois P. Heinz_, Sep 19 2013