A289842 Sum of products of terms in all partitions of 2*n into powers of 2.

1, 3, 11, 27, 83, 195, 515, 1155, 2899, 6387, 15219, 32883, 76275, 163059, 368883, 780531, 1738259, 3653715, 8022355, 16759635, 36428371, 75765843, 163217491, 338120787, 723384915, 1493913171, 3176799827, 6542573139, 13844246099, 28447592019, 59934789203

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..3309

Formula

a(n) = [x^(2*n)] Product_{k>=0} 1/(1 - 2^k*x^(2^k)). - Ilya Gutkovskiy, Sep 10 2018

a(n) ~ c * n * 2^n, where c = 2.1343755406794500897789546611306737041750472866941557748356... - Vaclav Kotesovec, Jun 18 2019

Example

n | partitions of 2*n into powers of 2                 | a(n)
--------------------------------------------------------------------------
1 | 2  , 1+1                                           | 2+1         =  3.
2 | 4  , 2+2  , 2+1+1, 1+1+1+1                         | 4+4+2+1     = 11.
3 | 4+2, 4+1+1, 2+2+2, 2+2+1+1, 2+1+1+1+1, 1+1+1+1+1+1 | 8+4+8+4+2+1 = 27.

Maple

b:= proc(n, i, p) option remember; `if`(n=0, p,
     `if`(i<1, 0, add(b(n-j*i, i/2, p*i^j), j=0..n/i)))
    end:
a:= n-> (t-> b(t, 2^ilog2(t), 1))(2*n):
seq(a(n), n=0..33);  # Alois P. Heinz, Oct 27 2017

Mathematica

b[n_, i_, p_] := b[n, i, p] = If[n == 0, p, If[i < 1, 0, Sum[b[n - j i, i/2, p i^j], {j, 0, n/i}]]];
a[n_] := b[2n, 2^Floor@Log[2, 2n], 1];
a /@ Range[0, 33] (* Jean-François Alcover, Nov 23 2020, after Alois P. Heinz *)

Crossrefs

Cf. A000123, A018819.

Sequence in context: A059400 A250223 A250271 * A077776 A113836 A036571

Adjacent sequences: A289839 A289840 A289841 * A289843 A289844 A289845

Keyword

nonn

Author

Seiichi Manyama, Oct 27 2017

Status

approved