A304393 Expansion of Product_{k>0} (1 + Sum_{m>=0} x^(k*2^m)).

1, 1, 2, 2, 5, 5, 8, 10, 17, 19, 27, 33, 48, 56, 76, 92, 126, 146, 192, 228, 298, 352, 444, 528, 667, 783, 969, 1145, 1414, 1658, 2017, 2365, 2878, 3352, 4027, 4703, 5634, 6548, 7773, 9033, 10705, 12381, 14573, 16857, 19790, 22800, 26631, 30655, 35723, 41005

Offset

0,3

Comments

Also the number of partitions of n in which each part occurs a power of 2 (cf. A000079) of times.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Seiichi Manyama)

Example

n | Partitions of n in which each part occurs a power of 2 (cf. A000079) of times
--+------------------------------------------------------------------------------
1 | 1;
2 | 2 = 1+1;
3 | 3 = 2+1;
4 | 4 = 3+1 = 2+2 = 2+1+1 = 1+1+1+1;
5 | 5 = 4+1 = 3+2 = 3+1+1 = 2+2+1;
6 | 6 = 5+1 = 4+2 = 4+1+1 = 3+2+1 = 3+3 = 2+2+1+1 = 2+1+1+1+1;
7 | 7 = 6+1 = 5+2 = 5+1+1 = 4+3 = 4+2+1 = 3+3+1 = 3+2+2 = 3+2+1+1 = 3+1+1+1+1;

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+add(b(n-i*2^j, i-1), j=0..ilog2(n/i))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, May 13 2018

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0,
   b[n, i-1] + Sum[b[n-i*2^j, i-1], {j, 0, Floor@Log2[n/i]}]]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Feb 14 2023, after Alois P. Heinz *)

Crossrefs

Cf. A000079, A055922, A300446, A304332.

Sequence in context: A183563 A222706 A240495 * A325535 A345165 A062405

Adjacent sequences: A304390 A304391 A304392 * A304394 A304395 A304396

Keyword

nonn

Author

Seiichi Manyama, May 12 2018

Status

approved