A225860 - OEIS

%I #20 Oct 18 2014 09:17:21

%S 1,1,3,3,10,16,55,133,599,1956,11982,57872,477289,3290993,37671322,

%T 373566217,5986589127,85738839408,1931359427404,40003346563574,

%U 1274368885871702,38222180804346119,1729302096638372691,75195441157176495562,4848355840082055530710

%N Number of partitions of 2^n into binomial coefficients C(n,k).

%e n=3: C(3,0)=C(3,3) = 1, C(3,1)=C(3,2) = 3, 2^3 = 8:

%e a(3) = #{3+3+1+1, 3+1+1+1+1+1, 8x1} = 3;

%e n=4: C(4,0)=C(4,4) = 1, C(4,1)=C(4,3) = 4, C(4,2) = 6, 2^4 = 16:

%e a(4) = #{6+6+4, 6+6+1+1+1+1, 6+4+4+1+1, 6+4+6x1, 6+10x1, 4+4+4+4, 4+4+4+1+1+1+1, 4+4+8x1, 4+12x1, 16x1} = 10

%e n=5: C(5,0)=C(5,5) = 1, C(5,1)=C(5,4) = 5, C(5,2)=C(5,3) = 10, 2^5 = 32:

%e a(5) = #{10+10+10+1+1, 10+10+5+5+1+1, 10+10+5+7x1, 10+10+12x1, 10+5+5+5+5+1+1, 10+5+5+5+7x1, 10+5+5+12x1, 10+5+17x1, 10+22x1, 6x5, 5x5+7x1, 5+5+5+5+12x1, 5+5+5+17x1, 5+5+22x1, 5+27x1, 32x1} = 16.

%p a:= proc(n) option remember; local g, i, j, l, m, t;

%p       m:= 1+iquo(n, 2);

%p       l:= array(1..m, [seq(binomial(n,k), k=0..m-1)]);

%p       g:= array(1..m, [seq(array(0..l[i]-1, [0$(l[i])]), i=1..m)]);

%p       g[1][0]:= 1;

%p       for j from 0 to 2^n do for i from 2 to m do

%p         g[i][irem(j, l[i])]:= g[i][irem(j, l[i])]

%p                              +g[i-1][irem(j, l[i-1])]

%p       od od; g[m][irem(2^n, l[m])]

%p     end:

%p seq(a(n), n=0..14);  # _Alois P. Heinz_, May 30 2013

%o (Haskell)

%o a225860 n = p (a034868_row n) (2 ^ n) where

%o    p _          0 = 1

%o    p []         _ = 0

%o    p bs'@(b:bs) m = if m < b then 0 else p bs' (m - b) + p bs m

%Y Cf. A000079, A034868, A007318.

%K nonn

%O 0,3

%A _Reinhard Zumkeller_, May 26 2013

%E a(13)-a(23) from _Alois P. Heinz_, May 30 2013

%E a(24) from _Alois P. Heinz_, Oct 06 2014