A280451 G.f.: Product_{k>=1, j>=1} (1+x^(j*k^2)).

1, 1, 1, 2, 3, 4, 5, 7, 9, 13, 16, 20, 27, 34, 42, 53, 67, 82, 102, 125, 153, 188, 227, 274, 332, 401, 478, 574, 686, 815, 969, 1147, 1356, 1600, 1884, 2210, 2597, 3040, 3547, 4141, 4824, 5607, 6508, 7546, 8732, 10100, 11656, 13431, 15473, 17793, 20429, 23436

Offset: 0

Author

Vaclav Kotesovec, Jan 03 2017

Keywords

• nonn

Links

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

Crossrefs

Cf. A004101, A006171, A107742, A280663, A280664.

Programs

Mathematica

nmax = 100; CoefficientList[Series[Product[(1 + x^(j*k^2)), {k, 1, Floor[Sqrt[nmax]+1]}, {j, 1, Floor[nmax/k^2] + 1}], {x, 0, nmax}], x]

PARI

my(N=66, x='x+O('x^N)); Vec(prod(k=1, sqrt(N), eta(x^(2*k^2))/eta(x^(k^2)))) \\ Seiichi Manyama, Apr 29 2021

Formula

a(n) ~ exp(Pi^2*sqrt(n/2)/3 + sqrt(3) * (sqrt(2)-1) * Zeta(1/2) * Zeta(3/2) * n^(1/4) / (2^(3/4) * sqrt(Pi)) - 9*((sqrt(2)-1) * Zeta(1/2) * Zeta(3/2))^2 / (16*Pi^3)) * sqrt(Pi) / (2^(3/2) * sqrt(3) * n^(3/4)).