A291749 Expansion of Product_{k>=1} (1 + x^(2*k^2 + 1)).

1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 1, 1, 0

Offset

0,74

Comments

In general, if d > 0, c > 0, GCD(d,c) = 1 and g.f. is Product_{k>=1} (1 + x^(d*k^2 + c)), then a(n) ~ ((sqrt(2)-1) * zeta(3/2))^(1/3) * exp(3*Pi^(1/3) * ((sqrt(2)-1) * zeta(3/2))^(2/3) * n^(1/3) / (2^(5/3) * d^(1/3))) / (2^(4/3) * Pi^(1/3) * sqrt(3) * d^(1/6) * n^(5/6)) * (1 + c * Pi^(2/3) * (sqrt(2)-1)^(4/3) * zeta(1/2) * zeta(3/2)^(1/3) / (2^(11/6) * d^(2/3) * n^(1/3))). - Vaclav Kotesovec, Mar 14 2026

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..100000

Formula

a(n) ~ exp(3 * Pi^(1/3) * ((sqrt(2)-1) * Zeta(3/2))^(2/3) * n^(1/3)/4) * ((sqrt(2)-1) * Zeta(3/2))^(1/3) / (2 * sqrt(6) * Pi^(1/3) * n^(5/6)).

Mathematica

nmax = 200; CoefficientList[Series[Product[(1 + x^(2*k^2 + 1)), {k, 1, Sqrt[(nmax - 1)/2] + 1}], {x, 0, nmax}], x]
nmax = 200; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[4]] = 1; Do[Do[poly[[j + 1]] += poly[[j - 2*k^2]], {j, nmax, 2*k^2 + 1, -1}]; , {k, 2, Sqrt[(nmax - 1)/2] + 1}]; poly

Crossrefs

Cf. A033461, A291748.

Sequence in context: A007949 A191265 A320003 * A370599 A253786 A078595

Adjacent sequences: A291746 A291747 A291748 * A291750 A291751 A291752

Keyword

nonn

Author

Vaclav Kotesovec, Aug 31 2017

Status

approved