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A259401 a(n) = Sum_{k=0..n} 2^(n-k)*p(k), where p(k) is the partition function A000041. 4
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 (list; graph; refs; listen; history; text; internal format)
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: A099050 A065352 A161993 * A008466 A102712 A054480

Adjacent sequences:  A259398 A259399 A259400 * A259402 A259403 A259404

KEYWORD

nonn

AUTHOR

Vaclav Kotesovec, Jun 26 2015

STATUS

approved

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Last modified July 6 07:41 EDT 2020. Contains 335476 sequences. (Running on oeis4.)