A394204 Expansion of Product_{k>=1} (1 + x^(k*(2*k-1))) / (1 - x^(k*(2*k-1))).

1, 2, 2, 2, 2, 2, 4, 6, 6, 6, 6, 6, 8, 10, 10, 12, 14, 14, 16, 18, 18, 22, 26, 26, 28, 30, 30, 34, 40, 42, 46, 50, 50, 54, 62, 66, 72, 78, 78, 82, 90, 94, 100, 110, 114, 122, 134, 138, 144, 158, 166, 178, 194, 198, 204, 218, 228, 242, 262, 270, 282, 302, 314, 330

Offset

0,2

Comments

Convolution of A279279 and A278949.

Links

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

Formula

a(n) ~ Gamma(1 + b/d) * ((4-sqrt(2))*zeta(3/2))^(2/3 + b/(3*d)) * d^(1/6 + b/(3*d)) * exp(3*Pi^(1/3) * ((4-sqrt(2))*zeta(3/2))^(2/3) * (n/d)^(1/3) / 4) / (2^(7/2 + 3*b/(2*d)) * sqrt(3) * Pi^(7/6 - b/(6*d)) * n^(7/6 + b/(3*d))), where d = 2, b = -1.

Mathematica

nmax = 100; CoefficientList[Series[Product[(1 + x^(k*(2*k-1))) / (1 - x^(k*(2*k-1))), {k, 1, Floor[Sqrt[1 + 8*nmax]/4 + 1]}], {x, 0, nmax}], x]

Crossrefs

Cf. A103265, A279279, A278949, A394202, A394203.

Sequence in context: A327817 A353643 A168260 * A008737 A244460 A160419

Adjacent sequences: A394201 A394202 A394203 * A394205 A394206 A394207

Keyword

nonn

Author

Vaclav Kotesovec, Mar 12 2026

Status

approved