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A259399
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a(n) = Sum_{k=0..n} p(k)^2, where p(k) is the partition function A000041.
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6
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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
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OFFSET
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0,2
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COMMENTS
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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.
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 0..5000
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FORMULA
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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
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MAPLE
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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
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MATHEMATICA
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Table[Sum[PartitionsP[k]^2, {k, 0, n}], {n, 0, 50}]
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CROSSREFS
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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
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KEYWORD
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nonn
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AUTHOR
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Vaclav Kotesovec, Jun 26 2015
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STATUS
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approved
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