A376631 G.f.: Sum_{k>=0} x^(k*(k+1)/2) * Product_{j=1..k} (1 + x^(2*j)).

1, 1, 0, 2, 0, 1, 1, 1, 1, 1, 2, 0, 3, 0, 2, 1, 3, 1, 3, 1, 2, 3, 2, 3, 2, 4, 1, 5, 2, 5, 2, 6, 1, 7, 2, 7, 3, 6, 4, 7, 5, 6, 7, 6, 7, 7, 9, 5, 11, 5, 12, 6, 14, 5, 15, 6, 16, 7, 17, 7, 18, 9, 18, 11, 19, 12, 20, 14, 19, 17, 19, 19, 20, 23, 18, 27, 18, 29, 20, 32, 19

Offset: 0

Author

Vaclav Kotesovec, Sep 30 2024

Keywords

• nonn

Links

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

Crossrefs

Cf. A000009, A053258, A053261, A264905, A333179, A376580, A376632, A376660.

Programs

Mathematica

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

Formula

G.f.: Sum_{k>=0} Product_{j=1..k} (x^j + x^(3*j)).

a(n) ~ c * A376660^sqrt(n) / sqrt(n), where c = 1/(2*sqrt(3 - 4*sinh(arcsinh(3^(3/2)/2) / 3) / sqrt(3))) = 0.39098976711379944962936707496887239986756106886318...

a(n) ~ A376580(n) * (A376660/A376621)^sqrt(n).