A259401 a(n) = Sum_{k=0..n} 2^(n-k)*p(k), where p(k) is the partition function A000041.

1, 3, 8, 19, 43, 93, 197, 409, 840, 1710, 3462, 6980, 14037, 28175, 56485, 113146, 226523, 453343, 907071, 1814632, 3629891, 7260574, 14522150, 29045555, 58092685, 116187328, 232377092, 464757194, 929518106, 1859040777, 3718087158, 7436181158, 14872370665

Offset: 0

Author

Vaclav Kotesovec, Jun 26 2015

Keywords

• nonn

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

Crossrefs

Cf. A000041, A048651, A090764, A259400, A292746.

Programs

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}]

PARI

a(n) = sum(k=0, n, 2^(n-k)*numbpart(k)); \\ Michel Marcus, Dec 03 2019

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