Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #16 Jun 24 2022 19:58:29
%S 1,2,6,15,40,89,210,435,919,1819,3583,6719,12648,22849,41074,72050,
%T 125411,213620,361845,601945,995074,1622338,2626342,4201367,6681992,
%U 10515756,16449852,25509952,39333476,60172701,91577517,138390481,208096282,310976731,462512831
%N a(n) = Sum_{k=0..n} p(k)^2, where p(k) is the partition function A000041.
%C 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.
%H Alois P. Heinz, <a href="/A259399/b259399.txt">Table of n, a(n) for n = 0..5000</a>
%F a(n) ~ exp(2*Pi*sqrt(2*n/3)) / (16*sqrt(6)*Pi*n^(3/2)).
%F a(n) = 1 + A209536(n). - _Alois P. Heinz_, Oct 21 2018
%p a:= proc(n) option remember; `if`(n<0, 0,
%p combinat[numbpart](n)^2+a(n-1))
%p end:
%p seq(a(n), n=0..40); # _Alois P. Heinz_, Oct 21 2018
%t Table[Sum[PartitionsP[k]^2,{k,0,n}],{n,0,50}]
%Y Cf. A000041, A000070, A209536, A265093.
%Y Partial sums of A001255.
%K nonn
%O 0,2
%A _Vaclav Kotesovec_, Jun 26 2015