A265093 - OEIS

%I #5 Feb 01 2016 10:05:53

%S 1,2,3,7,11,20,36,61,97,161,261,405,630,954,1438,2167,3191,4635,6751,

%T 9667,13763,19539,27460,38276,53160,73324,100549,137413,186697,252233,

%U 339849,455449,607549,808253,1070397,1412622,1858846,2436446,3182942,4147266

%N a(n) = Sum_{k=0..n} q(k)^2, where q(k) = partition numbers into distinct parts (A000009).

%C In general, for m >= 1, Sum_{k=0..n} q(k)^m ~ 2*sqrt(3*n)/(m*Pi) * q(n)^m ~ exp(Pi*m*sqrt(n/3)) / (Pi*m * 2^(2*m-1) * 3^(m/4-1/2) * n^(3*m/4-1/2)), where q(k) is A000009(k).

%H Alois P. Heinz, <a href="/A265093/b265093.txt">Table of n, a(n) for n = 0..10000</a>

%F a(n) = Sum_{k=0..n} A000009(k)^2.

%F a(n) ~ exp(2*Pi*sqrt(n/3))/(16*Pi*n).

%t Table[Sum[PartitionsQ[k]^2, {k,0,n}], {n,0,50}]

%Y Cf. A000070, A036469, A259399.

%K nonn

%O 0,2

%A _Vaclav Kotesovec_, Dec 01 2015