A333151 G.f.: Sum_{k>=1} (k^3 * x^(k^2) / Product_{j=1..k} (1 - x^j)).

0, 1, 1, 1, 9, 9, 17, 17, 25, 52, 60, 87, 122, 149, 184, 238, 337, 391, 517, 635, 825, 970, 1224, 1433, 1778, 2176, 2585, 3074, 3736, 4414, 5292, 6223, 7354, 8626, 10135, 11785, 13915, 16068, 18701, 21600, 25141, 28884, 33512, 38288, 44165, 50494, 57961

Offset: 0

Author

Vaclav Kotesovec, Mar 09 2020

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A003114, A268188, A333141.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i))))
    end:
a:= n-> add(k^3 * b(n-k^2, k), k=1..floor(sqrt(n))):
seq(a(n), n=0..50);  # after Alois P. Heinz

Mathematica

nmax = 50; CoefficientList[Series[Sum[n^3*x^(n^2)/Product[1-x^k, {k, 1, n}], {n, 0, Sqrt[nmax]}], {x, 0, nmax}], x]

Formula

a(n) ~ c * exp(2*Pi*sqrt(n/15)) * n^(3/4), where c = A333155^3 * phi^(1/2) / (2 * 3^(1/4) * 5^(1/2)) = 0.04512265567211918167849606290245... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio.