A248377 Number of compositions of 1 into parts 1/2^k with 0 <= k <= n.

1, 2, 6, 56, 5272, 47350056, 3820809588459176, 24878564279781563409541239097464, 1054787931172699885204409659788147413348784265452313995416385160

Offset: 0

Author

Bassam Abdul-Baki, Oct 05 2014

Keywords

• nonn

Comments

Equivalently, the number of compositions of 2^n into powers of 2.

Examples

			a(0) = 1: [1].
a(1) = 2: [1/2,1/2], [1].
a(2) = 6: [1/4,1/4,1/4,1/4], [1/2,1/4,1/4], [1/4,1/2,1/4], [1/4,1/4,1/2],  [1/2,1/2], [1].
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..11
• B. Abdul-Baki, Rhythmic Figures

Crossrefs

Cf. A023359.

Row sums of A323840.

Programs

Maple

b:= proc(n) option remember; `if`(n=0, 1,
      add(b(n-2^j), j=0..ilog2(n)))
    end:
a:= n-> b(2^n):
seq(a(n), n=0..10);  # Alois P. Heinz, Oct 20 2014

Mathematica

$RecursionLimit = 2000; Clear[b]; b[n_] := b[n] = If[n == 0, 1, Sum[b[n - 2^j], {j, 0, Log[2, n] // Floor}]]; a[n_] := b[2^n]; Table[a[n], {n, 0, 10}] (* Jean-François Alcover, Oct 30 2014, after Alois P. Heinz *)

Formula

a(n) = A023359(2^n).

lim_{n->oo} a(n+1)/a(n)^2 = 1.704176310706592045608982.... - Bassam Abdul-Baki, Sep 03 2020