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%I #14 Nov 28 2015 16:50:49
%S 1,2,6,18,42,102,238,522,1130,2394,4926,9978,19890,38942,75254,143598,
%T 270506,504126,929926,1698322,3074010,5516898,9820550,17349554,
%U 30430610,53007162,91734262,157771538,269734714,458542822,775281982,1303971722,2182227546,3634444634
%N O.g.f.: exp( Sum_{n>=1} (sigma(2*n^2)-sigma(n^2)) * x^n/n ).
%C Compare g.f. to the formula for Jacobi theta_4(x) given by:
%C _ theta_4(x) = exp( Sum_{n>=1} -(sigma(2*n)-sigma(n))*x^n/n )
%C where theta_4(x) = 1 + Sum_{n>=1} 2*(-x)^(n^2).
%C Here sigma(n) = A000203(n) is the sum of divisors of n.
%H Vaclav Kotesovec, <a href="/A195584/b195584.txt">Table of n, a(n) for n = 0..5000</a>
%F O.g.f.: exp( Sum_{n>=1} A054785(n^2)*x^n/n ), where exp( Sum_{n>=1} A054785(n)*x^n/n ) = 1/(1+2*Sum_{n>=1} (-x)^(n^2)), which is the g.f. of A015128.
%e G.f.: A(x) = 1 + 2*x + 6*x^2 + 18*x^3 + 42*x^4 + 102*x^5 + 238*x^6 +...
%e where
%e log(A(x)) = 2*x + 8*x^2/2 + 26*x^3/3 + 32*x^4/4 + 62*x^5/5 + 104*x^6/6 +...+ A195585(n)*x^n/n +...
%t nmax = 40; CoefficientList[Series[Exp[Sum[(DivisorSigma[1, 2*n^2] - DivisorSigma[1, n^2])*(x^n/n), {n, 1, nmax}]], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Nov 28 2015 *)
%o (PARI) {a(n)=polcoeff(exp(sum(m=1, n, (sigma(2*m^2)-sigma(m^2))*x^m/m)+x*O(x^n)), n)}
%Y Cf. A195585, A215603, A177399, A015128, A054785; variant: A225958.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Sep 20 2011