A259399 a(n) = Sum_{k=0..n} p(k)^2, where p(k) is the partition function A000041.

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

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.

Partial sums of A001255.

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