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A216406 G.f.: Product_{n>=1} ((1-x^n)/(1+x^n))^(2*n).

%I A216406 #20 May 02 2017 10:58:00
%S A216406 1,-4,0,8,16,-8,-48,-56,0,116,256,264,-32,-648,-1296,-1392,-352,2040,
%T A216406 5200,7368,6112,-784,-13744,-29304,-39648,-33804,-1376,60368,139552,
%U A216406 205304,210208,103432,-146528,-521744,-928480,-1190000,-1069904,-339720,1110864,3146640,5278624
%N A216406 G.f.: Product_{n>=1} ((1-x^n)/(1+x^n))^(2*n).
%C A216406 The number of contiguous signs seems to increase in proportion to the square-root of the number of terms.
%C A216406 Compare the g.f. to the Jacobi theta_4 series identity:
%C A216406 exp( Sum_{n>=1} -(sigma(2*n) - sigma(n))*x^n/n ) = 1 + 2*Sum_{n>=1} (-x)^(n^2).
%H A216406 Seiichi Manyama, <a href="/A216406/b216406.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..2500 from Paul D. Hanna)
%F A216406 G.f.: exp( Sum_{n>=1} -(sigma_2(2*n) - sigma_2(n))*x^n/n ) where sigma_2(n) = sum of squares of divisors of n.
%e A216406 G.f.: A(x) = 1 - 4*x + 8*x^3 + 16*x^4 - 8*x^5 - 48*x^6 - 56*x^7 + 116*x^9 +...
%e A216406 where the g.f. equals the infinite product:
%e A216406 A(x) = (1-x)^2/(1+x)^2 * (1-x^2)^4/(1+x^2)^4 * (1-x^3)^6/(1+x^3)^6 * (1-x^4)^8/(1+x^4)^8 * (1-x^5)^10/(1+x^5)^10 *...
%e A216406 The logarithm of the g.f. is illustrated by:
%e A216406 -log(A(x)) = 4*x + 16*x^2/2 + 40*x^3/3 + 64*x^4/4 + 104*x^5/5 + 160*x^6/6 + 200*x^7/7 + 256*x^8/8 +...+ 4*A076577(n)*x^n/n +...
%o A216406 (PARI) {a(n)=polcoeff(exp(sum(m=1, n+1, -(sigma(2*m,2)-sigma(m,2))*x^m/m+x*O(x^n))), n)}
%o A216406 (PARI) {a(n)=polcoeff(prod(m=1,n,((1-x^m)/(1+x^m +x*O(x^n)))^(2*m)), n)}
%o A216406 for(n=0, 100, print1(a(n), ", "))
%Y A216406 Cf. A156616, A076577, A001157 (sigma_2), A261386.
%K A216406 sign
%O A216406 0,2
%A A216406 _Paul D. Hanna_, Sep 06 2012