A308297 Expansion of Sum_{k>=1} mu(k)*log((theta_3(x^k) + 1)/2)/k, where theta_3() is the Jacobi theta function.

1, -1, 0, 1, -1, 1, -1, 0, 1, -1, 2, -3, 1, 1, -2, 5, -6, 4, -2, -3, 10, -15, 15, -9, -1, 17, -34, 43, -39, 17, 25, -78, 117, -127, 93, 3, -147, 298, -394, 369, -168, -211, 680, -1092, 1251, -939, 39, 1336, -2827, 3855, -3715, 1857, 1777, -6529, 10922, -12789, 9929

Offset: 1

Author

Ilya Gutkovskiy, May 19 2019

Keywords

• sign

Comments

Inverse Euler transform of A010052.

Crossrefs

Cf. A000122, A001156, A008683, A010052, A316152.

Programs

Mathematica

nmax = 57; CoefficientList[Series[Sum[MoebiusMu[k] Log[(EllipticTheta[3, 0, x^k] + 1)/2]/k, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
nmax = 57; CoefficientList[Series[Sum[MoebiusMu[k] Log[Sum[x^(j^2 k), {j, 0, nmax}]]/k, {k, 1, nmax}], {x, 0, nmax}], x] // Rest

Formula

G.f.: Sum_{k>=1} mu(k)*log(Sum_{j>=0} x^(j^2*k))/k.

Product_{n>=1} 1/(1 - x^n)^a(n) = g.f. of A010052.