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A354647 G.f. A(x) satisfies: -x^2 = Sum_{n=-oo..oo} (-1)^n * x^(n*(n+1)/2) * A(x)^(n*(n-1)/2).

%I A354647 #9 Jun 25 2022 10:04:47
%S A354647 1,0,1,3,9,25,78,256,881,3064,10831,38766,140550,514625,1900301,
%T A354647 7067013,26448613,99539716,376489459,1430330451,5455742957,
%U A354647 20885223619,80213926069,309002022843,1193616950854,4622372591972,17942238661229,69795082381496,272046051362013
%N A354647 G.f. A(x) satisfies: -x^2 = Sum_{n=-oo..oo} (-1)^n * x^(n*(n+1)/2) * A(x)^(n*(n-1)/2).
%F A354647 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
%F A354647 (1) -x^2 = Sum_{n=-oo..oo} (-1)^n * x^(n*(n-1)/2) * A(x)^(n*(n+1)/2).
%F A354647 (2) -x^2 = Sum_{n>=0} (-1)^n * x^(n*(n-1)/2) * (1 - x^(2*n+1)) * A(x)^(n*(n+1)/2).
%F A354647 (3) -x^2 = Sum_{n>=0} (-1)^n * A(x)^(n*(n-1)/2) * (1 - A(x)^(2*n+1)) * x^(n*(n+1)/2).
%F A354647 (4) -x^2 = Product_{n>=1} (1 - x^n*A(x)^n) * (1 - x^(n-1)*A(x)^n) * (1 - x^n*A(x)^(n-1)), by the Jacobi triple product identity.
%e A354647 G.f.: A(x) = 1 + x^2 + 3*x^3 + 9*x^4 + 25*x^5 + 78*x^6 + 256*x^7 + 881*x^8 + 3064*x^9 + 10831*x^10 + 38766*x^11 + 140550*x^12 + ...
%e A354647 such that A = A(x) satisfies:
%e A354647 (1) -x^2 = ... + x^36*A^28 - x^28*A^21 + x^21*A^15 - x^15*A^10 + x^10*A^6 - x^6*A^3 + x^3*A - x + 1 - A + x*A^3 - x^3*A^6 + x^6*A^10 - x^10*A^15 + x^15*A^21 - x^21*A^28 + x^28*A^36 + ...
%e A354647 (2) -x^2 = (1-x) - (1-x^3)*A + x*(1-x^5)*A^3 - x^3*(1-x^7)*A^6 + x^6*(1-x^9)*A^10 - x^10*(1-x^11)*A^15 + x^15*(1-x^13)*A^21 - x^21*(1-x^15)*A^28 + ...
%e A354647 (3) -x^2 = (1-A) - (1-A^3)*x + A*(1-A^5)*x^3 - A^3*(1-A^7)*x^6 + A^6*(1-A^9)*x^10 - A^10*(1-A^11)*x^15 + A^15*(1-A^13)*x^21 - A^21*(1-A^15)*x^28 + ...
%e A354647 (4) -x^2 = (1 - x*A)*(1 - A)*(1-x) * (1 - x^2*A^2)*(1 - x*A^2)*(1 - x^2*A) * (1 - x^3*A^3)*(1 - x^2*A^3)*(1 - x^3*A^2) * (1 - x^4*A^4)*(1 - x^3*A^4)*(1 - x^4*A^3) * (1 - x^5*A^5)*(1 - x^4*A^5)*(1 - x^5*A^4) * ...
%o A354647 (PARI) {a(n) = my(A=[1]); for(i=1,n, A = concat(A,0);
%o A354647 A[#A] = polcoeff(x^2 + sum(m=0,sqrtint(2*#A+9), (-1)^m * x^(m*(m-1)/2) * (1 - x^(2*m+1)) * Ser(A)^(m*(m+1)/2) ),#A-1) );A[n+1]}
%o A354647 for(n=0,30,print1(a(n),", "))
%Y A354647 Cf. A268650, A354648, A354649.
%K A354647 nonn
%O A354647 0,4
%A A354647 _Paul D. Hanna_, Jun 21 2022