%I #8 Jun 26 2019 04:02:44
%S 1,4,16,64,208,656,1984,5632,15520,41476,107312,271232,670464,1622160,
%T 3854208,9003264,20696640,46895248,104827472,231353984,504592448,
%U 1088323584,2322683072,4908033280,10273819136,21313971876,43843093488
%N O.g.f.: exp( Sum_{n>=1} (sigma(2n)-sigma(n))^2 * x^n/n ).
%C Here sigma(n) = A000203(n) is the sum of divisors of n. Compare g.f. to the formula for Jacobi theta_4(x) given by:
%C . theta_4(x) = exp( Sum_{n>=1} (sigma(n)-sigma(2n))*x^n/n )
%C where theta_4(x) = 1 + Sum_{n>=1} 2*(-x)^(n^2).
%H Vaclav Kotesovec, <a href="/A177398/b177398.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..400 from Paul D. Hanna)
%e G.f.: A(x) = 1 + 4*x + 16*x^2 + 64*x^3 + 208*x^4 + 656*x^5 +...
%e log(A(x)) = 4*x + 16*x^2/2 + 64*x^3/3 +...+ A054785(n)^2*x^n/n +...
%t nmax = 30; CoefficientList[Series[Exp[Sum[(DivisorSigma[1, 2*k] - DivisorSigma[1,k])^2 * x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jun 26 2019 *)
%o (PARI) {a(n)=polcoeff(exp(sum(m=1,n,(sigma(2*m)-sigma(m))^2*x^m/m)+x*O(x^n)),n)}
%Y Cf. A054785, A000203, A177399.
%K nonn
%O 0,2
%A _Paul D. Hanna_, May 30 2010