A225860 Number of partitions of 2^n into binomial coefficients C(n,k).

1, 1, 3, 3, 10, 16, 55, 133, 599, 1956, 11982, 57872, 477289, 3290993, 37671322, 373566217, 5986589127, 85738839408, 1931359427404, 40003346563574, 1274368885871702, 38222180804346119, 1729302096638372691, 75195441157176495562, 4848355840082055530710

Offset: 0

Author

Reinhard Zumkeller, May 26 2013

Keywords

• nonn

Examples

			n=3: C(3,0)=C(3,3) = 1, C(3,1)=C(3,2) = 3, 2^3 = 8:
a(3) = #{3+3+1+1, 3+1+1+1+1+1, 8x1} = 3;
n=4: C(4,0)=C(4,4) = 1, C(4,1)=C(4,3) = 4, C(4,2) = 6, 2^4 = 16:
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
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:
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.
		

Crossrefs

Cf. A000079, A034868, A007318.

Programs

Haskell

a225860 n = p (a034868_row n) (2 ^ n) where
   p _          0 = 1
   p []         _ = 0
   p bs'@(b:bs) m = if m < b then 0 else p bs' (m - b) + p bs m

Maple

a:= proc(n) option remember; local g, i, j, l, m, t;
      m:= 1+iquo(n, 2);
      l:= array(1..m, [seq(binomial(n,k), k=0..m-1)]);
      g:= array(1..m, [seq(array(0..l[i]-1, [0$(l[i])]), i=1..m)]);
      g[1][0]:= 1;
      for j from 0 to 2^n do for i from 2 to m do
        g[i][irem(j, l[i])]:= g[i][irem(j, l[i])]
                             +g[i-1][irem(j, l[i-1])]
      od od; g[m][irem(2^n, l[m])]
    end:
seq(a(n), n=0..14);  # Alois P. Heinz, May 30 2013

Extensions

a(13)-a(23) from Alois P. Heinz, May 30 2013

a(24) from Alois P. Heinz, Oct 06 2014