OFFSET
0,2
COMMENTS
In general, Sum_{k=0..n} (m^(n-k) * p(k)) ~ m^n / QPochhammer[1/m, 1/m], for m > 1.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..3320
FORMULA
a(n) ~ c * 2^n, where c = 1/A048651 = 1/QPochhammer[1/2, 1/2] = 3.462746619455...
G.f.: (1/(1 - 2*x)) * Product_{k>=1} 1/(1 - x^k). - Ilya Gutkovskiy, Dec 03 2019
MAPLE
a:= proc(n) option remember; `if`(n<0, 0,
2*a(n-1)+combinat[numbpart](n))
end:
seq(a(n), n=0..32); # Alois P. Heinz, Dec 03 2019
MATHEMATICA
Table[Sum[2^(n-k)*PartitionsP[k], {k, 0, n}], {n, 0, 50}]
PROG
(PARI) a(n) = sum(k=0, n, 2^(n-k)*numbpart(k)); \\ Michel Marcus, Dec 03 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, Jun 26 2015
STATUS
approved