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
Keywords
Examples
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.
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..3309
Programs
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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 *)
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