A394205 Expansion of Product_{k>=1} (1 + x^(k*(3*k+1)/2)) / (1 - x^(k*(3*k+1)/2)).

1, 0, 2, 0, 2, 0, 2, 2, 2, 4, 2, 4, 2, 4, 4, 6, 6, 8, 6, 8, 6, 10, 10, 12, 14, 12, 16, 12, 20, 16, 24, 20, 26, 24, 26, 30, 30, 36, 34, 40, 40, 44, 48, 52, 54, 58, 58, 68, 66, 78, 78, 84, 88, 88, 102, 100, 116, 118, 126, 132, 132, 148, 150, 166, 174, 180, 194, 192

Offset

0,3

Comments

Convolution of A296238 and A296237.

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 = 3/2, b = 1/2.

Mathematica

nmax = 120; CoefficientList[Series[Product[(1 + x^(k*(3*k+1)/2)) / (1 - x^(k*(3*k+1)/2)), {k, 1, Floor[Sqrt[1 + 24*nmax]/6 + 1]}], {x, 0, nmax}], x]

Crossrefs

Cf. A103265, A296238, A296237.

Sequence in context: A161840 A355851 A140302 * A085341 A221474 A160812

Adjacent sequences: A394202 A394203 A394204 * A394206 A394207 A394208

Keyword

nonn

Author

Vaclav Kotesovec, Mar 12 2026

Status

approved