A394207 Expansion of Product_{k>=1} (1 + x^(k*(3*k-1)/2)) / (1 - x^(k*(3*k-1)/2)).

1, 2, 2, 2, 2, 4, 6, 6, 6, 6, 8, 10, 12, 14, 14, 16, 18, 22, 26, 26, 28, 30, 36, 42, 44, 48, 50, 58, 66, 70, 76, 78, 86, 94, 102, 114, 120, 130, 138, 150, 168, 178, 190, 198, 212, 232, 246, 266, 280, 298, 320, 340, 370, 390, 410, 432, 458, 498, 526, 554, 582, 614

Offset

0,2

Comments

Convolution of A218380 and A218379.

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 = 100; 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, A218380, A218379.

Sequence in context: A391211 A386235 A064877 * A008363 A261611 A097198

Adjacent sequences: A394204 A394205 A394206 * A394208 A394209 A394210

Keyword

nonn

Author

Vaclav Kotesovec, Mar 12 2026

Status

approved