A333141 G.f.: Sum_{k>=1} (k^2 * x^(k^2) / Product_{j=1..k} (1 - x^j)).

0, 1, 1, 1, 5, 5, 9, 9, 13, 22, 26, 35, 48, 57, 70, 88, 117, 135, 173, 207, 261, 304, 374, 433, 528, 628, 739, 864, 1032, 1198, 1416, 1639, 1914, 2212, 2569, 2949, 3433, 3920, 4511, 5150, 5925, 6732, 7720, 8736, 9969, 11284, 12823, 14444, 16395, 18457, 20836

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.

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^2 * 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^2 * 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^(1/4), where c = A333155^2 * phi^(1/2) / (2 * 3^(1/4) * 5^(1/2)) = 0.076061100391958657489521534823556... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio.