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%I #7 Oct 13 2012 11:42:59
%S 1,-1,1,-3,6,-17,43,-125,348,-1029,3020,-9116,27567,-84620,260949,
%T -812053,2539208,-7989121,25244540,-80136851,255325972,-816447638,
%U 2618870068,-8425244209,27176810469,-87879769383,284813417885,-925013053556,3010106492409,-9813119711706
%N G.f. satisfies: A(x) = 1 / Product_{n>=1} (1 + x^n*A(x)) * (1 + x^n/A(x)) * (1-x^n).
%F G.f. satisfies: A(x) = 1 / Sum_{n>=1} (A(x)^n + x^n/A(x)^n) * x^(n*(n-1)/2) due to the Jacobi triple product identity.
%e G.f.: A(x) = 1 - x + x^2 - 3*x^3 + 6*x^4 - 17*x^5 + 43*x^6 - 125*x^7 +...
%e such that
%e 1/A(x) = (1+x*A(x))*(1-x/A(x))*(1-x) * (1+x^2*A(x))*(1-x^2/A(x))*(1-x^2) * (1+x^3*A(x))*(1-x^3/A(x))*(1-x^3) * (1+x^4*A(x))*(1-x^4/A(x))*(1-x^4) *...
%e 1/A(x) = (A(x) + x/A(x)) + (A(x)^2 + x^2/A(x)^2)*x + (A(x)^3 + x^3/A(x)^3)*x^3 + (A(x)^4 + x^4/A(x)^4)*x^6 + (A(x)^5 + x^5/A(x)^5)*x^10 +...
%o (PARI) {a(n)=local(A=1-x);for(i=1,n,A=1/prod(m=1,n,(1+x^m/A)*(1+x^m*A)*(1-x^m)+x*O(x^n)));polcoeff(A,n)}
%o (PARI) {a(n)=local(A=1-x);for(i=1,n,A=1/2*(A+1/sum(m=1,sqrtint(8*n+1),(A^m+x^m/A^m)*x^(m*(m-1)/2)+x*O(x^n))));polcoeff(A,n)}
%o for(n=0,21,print1(a(n),", "))
%Y Cf. A190791.
%K sign
%O 0,4
%A _Paul D. Hanna_, Sep 18 2012
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