A377301 G.f.: Sum_{k>=1} k * x^(k*(4*k-3)) / (1 - x^k).

1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 4, 3, 1, 6, 1, 3, 4, 3, 1, 6, 1, 3, 4, 3, 1, 6, 1, 3, 4, 3, 1, 6, 1, 3, 4, 7, 1, 6, 1, 7, 4, 3, 1, 10, 1, 3, 4, 7, 1, 6, 1, 7, 4, 3, 1, 10, 1, 3, 4, 7, 1, 6, 1, 7, 4, 3, 1, 10, 6, 3, 4, 7, 1, 11

Offset: 1

Author

Vaclav Kotesovec, Oct 23 2024

Keywords

• nonn

Comments

In general, for d > 0, if g.f. = Sum_{k>=1} k * x^(k*(d*k - d + 2)/2) / (1 - x^k), then Sum_{k=1..n} a(k) ~ 2^(3/2) * n^(3/2) / (3*sqrt(d)).

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Crossrefs

Column 8 of A334466.

Programs

Mathematica

Table[Sum[If[n > 4*k*(k-1), k, 0], {k, Divisors[n]}], {n, 1, 100}]
nmax = 100; Rest[CoefficientList[Series[Sum[k*x^(k*(4*k - 3))/(1 - x^k), {k, 1, Sqrt[nmax/4] + 1}], {x, 0, nmax}], x]]

Formula

Sum_{k=1..n} a(k) ~ n^(3/2) / 3.