A394244 Expansion of Product_{k>=1} 1/(1 - x^(2*k^2 + 1)).

1, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 2, 0, 0, 3, 1, 0, 3, 1, 0, 3, 1, 0, 4, 2, 0, 4, 2, 0, 5, 2, 0, 6, 3, 1, 6, 3, 1, 7, 3, 1, 8, 4, 2, 8, 4, 2, 10, 5, 2, 11, 6, 3, 12, 6, 3, 14, 7, 3, 15, 8, 4, 17, 8, 4, 19, 10, 5, 20, 12, 6, 22, 13, 6, 24, 15, 7, 25, 17, 8

Offset

0,10

Links

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

Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Formula

a(n) ~ sqrt(c) * zeta(3/2)^(2/3) * exp(3 * Pi^(1/3) * zeta(3/2)^(2/3) * n^(1/3) / (2^(4/3) * d^(1/3))) / (2^(7/3) * sqrt(3) * d^(1/3) * Pi^(1/6) * sinh(Pi*sqrt(c/d)) * n^(7/6)) * (1 - (c * Pi^(2/3) * zeta(1/2) * zeta(3/2)^(1/3) / (2^(5/3) * d^(2/3)) + 17*d^(1/3) / (9*Pi^(1/3) * (2*zeta(3/2))^(2/3))) / n^(1/3)), where d = 2, c = 1.

Mathematica

nmax = 200; CoefficientList[Series[1/Product[(1 - x^(2*k^2 + 1)), {k, 1, Sqrt[(nmax + 1)/2] + 1}], {x, 0, nmax}], x]

Crossrefs

Cf. A291749, A393990, A394243.

Sequence in context: A341792 A341793 A341773 * A359288 A096142 A216921

Adjacent sequences: A394241 A394242 A394243 * A394245 A394246 A394247

Keyword

nonn

Author

Vaclav Kotesovec, Mar 13 2026

Status

approved