A115625 Number of partitions of {1,...,n} into block sizes a power of 2.

1, 1, 2, 4, 11, 31, 106, 372, 1500, 6220, 28696, 136016, 702802, 3727946, 21253324, 124231096, 772458367, 4918962479, 33061095118, 227303570524, 1639389365201, 12082068856285, 92951844299150, 729991789222972, 5960617085224012, 49637008114202876

Offset: 0

Author

Christian G. Bower, Jan 26 2006

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..604

Crossrefs

Cf. A018819, A115626.

Programs

Maple

with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i<1, 0, add(multinomial(n, n-i*j, i$j)/j!*
      b(n-i*j, i/2), j=0..n/i)))
    end:
a:= n-> b(n, 2^ilog2(n)):
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 17 2015
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add((j->
      a(n-j)*binomial(n-1, j-1))(2^i), i=0..ilog2(n)))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Apr 21 2023

Mathematica

multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[multinomial[n, Join[{n-i*j}, Array[i&, j]]]/j!*b[n - i*j, i/2], {j, 0, n/i}]]]; a[n_] := b[n, 2^Floor[Log[2, n]]]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Oct 29 2015, after Alois P. Heinz *)

Formula

E.g.f.: exp(x+x^2/2+x^4/4!+x^8/8!+x^16/16!+...).