A394261 Expansion of Product_{k>=1} (1 + x^(2*k^2+1)) / (1 - x^(2*k^2+1)).

1, 0, 0, 2, 0, 0, 2, 0, 0, 4, 0, 0, 6, 0, 0, 6, 0, 0, 8, 2, 0, 10, 4, 0, 10, 4, 0, 12, 8, 0, 14, 12, 0, 16, 12, 0, 20, 16, 2, 22, 20, 4, 26, 20, 4, 32, 24, 8, 34, 28, 12, 40, 32, 12, 48, 40, 16, 52, 44, 20, 62, 52, 20, 72, 64, 24, 80, 68, 28, 94, 80, 32, 104, 98

Offset

0,4

Comments

Convolution of A291749 and A394244.

Links

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

Formula

a(n) ~ ((4 - sqrt(2))*zeta(3/2))^(2/3) * exp(3*Pi^(1/3) * ((4 - sqrt(2)) * zeta(3/2))^(2/3) * n^(1/3) / 2^(7/3)) / (2^(23/6) * sqrt(3) * Pi^(1/6) * sinh(Pi/sqrt(2)) * n^(7/6)).

Mathematica

nmax = 150; CoefficientList[Series[Product[(1+x^(2*k^2+1))/(1-x^(2*k^2+1)), {k, 1, Sqrt[(nmax+1)/2]+1}], {x, 0, nmax}], x]

Crossrefs

Cf. A291749, A394244, A394262.

Sequence in context: A382487 A307985 A024164 * A138805 A316400 A061897

Adjacent sequences: A394258 A394259 A394260 * A394262 A394263 A394264

Keyword

nonn

Author

Vaclav Kotesovec, Mar 14 2026

Status

approved