A343820 Number of partitions of 2n into powers of 2: p1 <= p2 <= ... <= p_k such that p_i <= 1 + Sum_{j=1..i-1} p_j.

1, 1, 2, 3, 6, 8, 12, 15, 26, 32, 42, 50, 68, 80, 98, 113, 166, 192, 230, 262, 318, 360, 418, 468, 572, 640, 732, 812, 934, 1032, 1160, 1273, 1626, 1792, 2010, 2202, 2482, 2712, 3006, 3268, 3682, 4000, 4402, 4762, 5254, 5672, 6190, 6658, 7492, 8064, 8772, 9412

Offset: 0

Author

Alois P. Heinz, Apr 30 2021

Keywords

• nonn

Examples

			a(2) = 2: [1,1,1,1], [1,1,2].
a(3) = 3: [1,1,1,1,1,1], [1,1,1,1,2], [1,1,2,2].
a(4) = 6: [1,1,1,1,1,1,1,1], [1,1,1,1,1,1,2], [1,1,1,1,2,2], [1,1,2,2,2], [1,1,1,1,4], [1,1,2,4].
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A000123, A000225, A002449, A018819, A343756, A343944.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<0, 0, (p->
      `if`(p>n or p>n-p+1, 0, b(n-p, i)))(2^i)+b(n, i-1)))
    end:
a:= n-> b(2*n, ilog2(n)+1):
seq(a(n), n=0..80);

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 0, 0, Function[p, If[p > n || p > n - p + 1, 0, b[n - p, i]]][2^i] + b[n, i - 1]]];
a[n_] := b[2n, BitLength[n] + 1];
Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Feb 13 2023, after Alois P. Heinz *)

Formula

a(n) is odd <=> n in { A000225 }.

a(2^(n-1)) = A002449(n).