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

0,2

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

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

Sequence in context: A065352 A161993 A360489 * A008466 A102712 A054480

Adjacent sequences: A259398 A259399 A259400 * A259402 A259403 A259404

Keyword

nonn

Author

Vaclav Kotesovec, Jun 26 2015

Status

approved