A394201 Expansion of Product_{k>=1} (1 + x^(k*(k+5)/2)) / (1 - x^(k*(k+5)/2)).

1, 0, 0, 2, 0, 0, 2, 2, 0, 2, 4, 0, 4, 4, 2, 6, 4, 4, 8, 8, 4, 12, 12, 4, 16, 18, 8, 18, 26, 12, 22, 32, 20, 32, 36, 30, 42, 48, 36, 54, 68, 40, 70, 88, 56, 86, 108, 80, 100, 134, 106, 126, 166, 132, 164, 200, 162, 206, 248, 200, 248, 312, 248, 298, 380, 312, 360

Offset

0,4

Comments

Convolution of A393989 and A393988.

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 = 1/2, b = 5/2.

Mathematica

nmax = 100; CoefficientList[Series[Product[(1 + x^(k*(k+5)/2)) / (1 - x^(k*(k+5)/2)), {k, 1, Floor[Sqrt[25 + 8*nmax]/2 + 1]}], {x, 0, nmax}], x]

Crossrefs

Cf. A103265, A280366, A393989, A393988.

Sequence in context: A275966 A284059 A329767 * A356018 A107502 A230419

Adjacent sequences: A394198 A394199 A394200 * A394202 A394203 A394204

Keyword

nonn

Author

Vaclav Kotesovec, Mar 12 2026

Status

approved