%I #11 Dec 30 2013 04:19:31

%S 0,1,10,2780,6631556521,71669271794142235712392433,

%T 78417479379491793666843945562521255790293292270929676484784808001

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

%H Alois P. Heinz, <a href="/A227223/b227223.txt">Table of n, a(n) for n = 0..9</a>

%F a(n) = (2^n)! * [x^(2^n)] exp(Sum_{j=1..n} x^j/j!).

%F a(n) = A229223(2^n,n).

%e 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 G:= proc(n, k) option remember; local j; if k>n then G(n, n)

%p elif n=0 then 1 elif k<1 then 0 else G(n-k, k);

%p for j from k-1 to 1 by -1 do %*(n-j)/j +G(n-j, k) od; % fi

%p end:

%p a:= n-> G(2^n, n):

%p seq(a(n), n=0..7);

%t 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 Cf. A229223.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Sep 19 2013