A281688 Expansion of Sum_{i>=0} x^(2^i)/(1 - x^(2^i)) / Product_{j>=0} (1 - x^(2^j)).

1, 3, 5, 10, 14, 23, 29, 45, 55, 79, 93, 130, 150, 199, 225, 296, 332, 423, 469, 594, 654, 807, 881, 1085, 1179, 1423, 1537, 1850, 1990, 2355, 2521, 2983, 3185, 3719, 3957, 4618, 4902, 5655, 5985, 6909, 7299, 8343, 8793, 10050, 10574, 11979, 12577, 14260, 14952, 16823, 17609, 19818, 20718, 23155, 24169, 27033

Offset: 1

Author

Ilya Gutkovskiy, Jan 27 2017

Keywords

• nonn

Comments

Total number of parts in all partitions of n into powers of 2 (A000079).

Convolution of A001511 and A018819.

Examples

			a(4) = 10 because we have [4], [2, 2], [2, 1, 1], [1, 1, 1, 1] and 1 + 2 + 3 + 4 = 10.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..20000
• Index entries for related partition-counting sequences

Crossrefs

Cf. A000079, A001511, A018819, A343944.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<0, 0, (p->
      `if`(p>n, 0, (h-> h+[0, h[1]])(b(n-p, i))))(2^i)+b(n, i-1)))
    end:
a:= n-> b(n, ilog2(n))[2]:
seq(a(n), n=1..56);  # Alois P. Heinz, May 04 2021

Mathematica

Rest[CoefficientList[Series[Sum[x^2^i/(1 - x^2^i), {i, 0, 20}]/Product[1 - x^2^j, {j, 0, 20}], {x, 0, 56}], x]]

Formula

G.f.: Sum_{i>=0} x^(2^i)/(1 - x^(2^i)) / Product_{j>=0} (1 - x^(2^j)).