%I #13 Jan 19 2024 09:36:01
%S 1,2,0,4,-6,32,-88,376,-1376,5574,-22232,91548,-378736,1589304,
%T -6719040,28647592,-122933470,530755764,-2303432600,10043949684,
%U -43979901840,193309672224,-852599615912,3772221225128,-16737583019616,74461240879386
%N G.f. satisfies: A(x) = 1 + Sum_{n>=1} (A(x)^n + A(x)^-n) * x^(n^2).
%H Vaclav Kotesovec, <a href="/A190791/b190791.txt">Table of n, a(n) for n = 0..200</a>
%F G.f. satisfies: A(x) = Product_{n>=1} (1 - x^(2n))*(1 + x^(2n-1)*A(x))*(1 + x^(2n-1)/A(x)), due to the Jacobi triple product identity.
%F a(n) ~ (-1)^(n+1) * c * d^n / n^(3/2), where d = 4.73097028144959... and c = 0.1236197969613... . - _Vaclav Kotesovec_, Mar 02 2016
%e G.f.: A(x) = 1 + 2*x + 4*x^3 - 6*x^4 + 32*x^5 - 88*x^6 + 376*x^7 +...
%e The g.f. A(x) satisfies the series:
%e * A(x) = 1 + (A(x) + A(x)^-1)*x + (A(x)^2 + A(x)^-2)*x^4 + (A(x)^3 + A(x)^-3)*x^9 + (A(x)^4 + A(x)^-4)*x^16 +...
%e * A(x) = (1-x^2)*(1+x*A(x))*(1+x/A(x)) * (1-x^4)*(1+x^3*A(x))*(1+x^3/A(x)) * (1-x^6)*(1+x^5*A(x))*(1+x^5/A(x)) *...
%e which is a result due to the Jacobi triple product identity.
%t (* Calculation of constant d: *) -1/r /. FindRoot[{r^2 * QPochhammer[r^2] * QPochhammer[-1/(r*s), r^2] * QPochhammer[-s/r, r^2] / ((r + s)*(1 + r*s)) == 1, -1 + r/(r + s) + 1/(1 + r*s) + (QPolyGamma[0, Log[-1/(r*s)]/Log[r^2], r^2] - QPolyGamma[0, Log[-s/r]/Log[r^2], r^2])/Log[r^2] == 1}, {r, -1/4}, {s, 1/2}, WorkingPrecision -> 70] (* _Vaclav Kotesovec_, Jan 19 2024 *)
%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=1+sum(m=1,n,(A^m+A^-m+x*O(x^n))*x^(m^2)));polcoeff(A,n)}
%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=prod(m=1,n,(1-x^(2*m))*(1+A*x^(2*m-1))*(1+A^-1*x^(2*m-1)+x*O(x^n))));polcoeff(A,n)}
%K sign
%O 0,2
%A _Paul D. Hanna_, May 20 2011