A376580 G.f.: Sum_{k>=0} x^(k^2) * Product_{j=1..k} (1 + x^(2*j-1))^2.

1, 1, 2, 1, 1, 2, 1, 2, 4, 3, 3, 3, 3, 4, 4, 5, 5, 7, 9, 7, 8, 9, 9, 10, 11, 12, 13, 14, 15, 18, 17, 19, 24, 23, 25, 27, 28, 31, 32, 33, 37, 40, 42, 44, 47, 52, 54, 59, 62, 67, 75, 75, 80, 87, 90, 95, 102, 109, 114, 119, 127, 134, 142, 150, 159, 171, 178, 187, 199, 211

Offset

0,3

Links

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

Formula

a(n) ~ c * A376621^sqrt(n) / sqrt(n), where c = 1/(2*sqrt(3 - 4*sinh(arcsinh(3^(3/2)/2) / 3) / sqrt(3))) = 0.390989767113799449629...

a(n) ~ c * A376542(n), where c = (108 + 12*sqrt(93))^(1/3)/3 - 4/(108 + 12*sqrt(93))^(1/3) = 1.364655607... is the real root of the equation c*(4 + c^2) = 8.

a(n) ~ c * A369557(n), where c = A347178 = -sinh(log((-3*sqrt(3) + sqrt(31))/2)/3) / sqrt(3) = 0.3411639019... is the real root of the equation 2*c*(1 + 4*c^2) = 1.

a(n) ~ A376631(n) * (A376621/A376660)^sqrt(n).

Mathematica

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

Crossrefs

Cf. A053258, A216222, A306734, A369557, A376542, A376581, A376621.

Sequence in context: A131325 A193749 A049824 * A133087 A153919 A185286

Adjacent sequences: A376577 A376578 A376579 * A376581 A376582 A376583

Keyword

nonn

Author

Vaclav Kotesovec, Sep 29 2024

Status

approved