A376632 G.f.: Sum_{k>=0} x^(k^2) * Product_{j=1..k} (1 + x^(2*j)).

1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 2, 1, 1, 1, 1, 1, 1, 2, 0, 2, 1, 2, 1, 2, 1, 2, 2, 1, 2, 1, 3, 2, 3, 1, 3, 1, 3, 2, 3, 2, 3, 3, 2, 4, 3, 4, 2, 4, 2, 5, 3, 5, 2, 5, 3, 5, 4, 4, 5, 5, 5, 5, 6, 4, 7, 4, 7, 4, 8, 4, 8, 5, 8, 6, 8, 6, 9, 7, 8, 8, 8, 9, 8, 10

Offset: 0

Author

Vaclav Kotesovec, Sep 30 2024

Keywords

• nonn

Links

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

Crossrefs

Cf. A306734, A376627, A376631.

Programs

Mathematica

nmax = 100; CoefficientList[Series[Sum[x^(k^2)*Product[1+x^(2*j), {j, 1, k}], {k, 0, Sqrt[nmax]}], {x, 0, nmax}], x]
nmax = 100; p = 1; s = 1; Do[p = Expand[p*(1 + x^(2*k))*x^(2*k - 1)]; p = Take[p, Min[nmax + 1, Exponent[p, x] + 1, Length[p]]]; s += p;, {k, 1, Sqrt[nmax]}]; Take[CoefficientList[s, x], nmax + 1]

Formula

a(n) ~ sqrt(1 + 3/sqrt(5)) * exp(Pi*sqrt(n/30)) / (4*sqrt(n)).