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Expansion of Sum_{k>=1} mu(k)*log((theta_3(x^k) + 1)/2)/k, where theta_3() is the Jacobi theta function.
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%I #5 May 19 2019 20:40:59

%S 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,

%T -34,43,-39,17,25,-78,117,-127,93,3,-147,298,-394,369,-168,-211,680,

%U -1092,1251,-939,39,1336,-2827,3855,-3715,1857,1777,-6529,10922,-12789,9929

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

%C Inverse Euler transform of A010052.

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

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

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

%t 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

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

%K sign

%O 1,11

%A _Ilya Gutkovskiy_, May 19 2019