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A259399 a(n) = Sum_{k=0..n} p(k)^2, where p(k) is the partition function A000041. 6
1, 2, 6, 15, 40, 89, 210, 435, 919, 1819, 3583, 6719, 12648, 22849, 41074, 72050, 125411, 213620, 361845, 601945, 995074, 1622338, 2626342, 4201367, 6681992, 10515756, 16449852, 25509952, 39333476, 60172701, 91577517, 138390481, 208096282, 310976731, 462512831 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

In general, Sum_{k=0..n} p(k)^m ~ sqrt(6*n)/(m*Pi) * p(n)^m ~ exp(m*Pi*sqrt(2*n/3)) / (m * Pi * 3^((m-1)/2) * 2^(2*m-1/2) * n^(m-1/2)), for m >= 1.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..5000

FORMULA

a(n) ~ exp(2*Pi*sqrt(2*n/3)) / (16*sqrt(6)*Pi*n^(3/2)).

a(n) = 1 + A209536(n). - Alois P. Heinz, Oct 21 2018

MAPLE

a:= proc(n) option remember; `if`(n<0, 0,

      combinat[numbpart](n)^2+a(n-1))

    end:

seq(a(n), n=0..40);  # Alois P. Heinz, Oct 21 2018

MATHEMATICA

Table[Sum[PartitionsP[k]^2, {k, 0, n}], {n, 0, 50}]

CROSSREFS

Cf. A000041, A000070, A209536, A265093.

Sequence in context: A026270 A321646 A246563 * A307128 A172399 A001654

Adjacent sequences:  A259396 A259397 A259398 * A259400 A259401 A259402

KEYWORD

nonn

AUTHOR

Vaclav Kotesovec, Jun 26 2015

STATUS

approved

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Last modified August 9 01:14 EDT 2020. Contains 336310 sequences. (Running on oeis4.)