A265093 a(n) = Sum_{k=0..n} q(k)^2, where q(k) = partition numbers into distinct parts (A000009).

1, 2, 3, 7, 11, 20, 36, 61, 97, 161, 261, 405, 630, 954, 1438, 2167, 3191, 4635, 6751, 9667, 13763, 19539, 27460, 38276, 53160, 73324, 100549, 137413, 186697, 252233, 339849, 455449, 607549, 808253, 1070397, 1412622, 1858846, 2436446, 3182942, 4147266

Offset: 0

Author

Vaclav Kotesovec, Dec 01 2015

Keywords

• nonn

Comments

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).

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A000070, A036469, A259399.

Programs

Mathematica

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

Formula

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

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