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A357207 Coefficients in the power series A(x) such that: x*A(x)^3 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.

%I A357207 #7 Sep 19 2022 11:09:01
%S A357207 1,1,7,55,469,4307,41678,418872,4330275,45754091,491916135,5364166402,
%T A357207 59186372395,659556170091,7412556531714,83921355689635,
%U A357207 956228695216241,10957322339242547,126189988012692329,1459793848341094130,16955390069787782159,197653935181097885580
%N A357207 Coefficients in the power series A(x) such that: x*A(x)^3 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.
%F A357207 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following relations.
%F A357207 (1) x*A(x)^3 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.
%F A357207 (2) -x*A(x)^4 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) / A(x)^n.
%F A357207 (3) x*A(x)^3 = Product_{n>=1} (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)) * (1 - x^n), due to the Jacobi triple product identity.
%F A357207 (4) -x*A(x)^4 = Product_{n>=1} (1 - x^n/A(x)) * (1 - x^(n-1)*A(x)) * (1 - x^n), due to the Jacobi triple product identity.
%e A357207 G.f.: A(x) = 1 + x + 7*x^2 + 55*x^3 + 469*x^4 + 4307*x^5 + 41678*x^6 + 418872*x^7 + 4330275*x^8 + 45754091*x^9 + 491916135*x^10 + ...
%e A357207 where
%e A357207 x*A(x)^3 = ... - x^10/A(x)^5 + x^6/A(x)^4 - x^3/A(x)^3 + x/A(x)^2 - 1/A(x) + 1 - x*A(x) + x^3*A(x)^2 - x^6*A(x)^3 + x^10*A(x)^4 -+ ... + (-1)^n * x^(n*(n+1)/2) * A(x)^n + ...
%o A357207 (PARI) {a(n) = my(A=[1, 1], t); for(i=1, n, A=concat(A, 0); t = ceil(sqrt(2*n+9));
%o A357207 A[#A] = polcoeff( x*Ser(A)^3 - sum(m=-t, t, (-1)^m*x^(m*(m+1)/2)*Ser(A)^m ), #A-1)); A[n+1]}
%o A357207 for(n=0, 30, print1(a(n), ", "))
%Y A357207 Cf. A355361, A357206, A357208, A357209.
%K A357207 nonn
%O A357207 0,3
%A A357207 _Paul D. Hanna_, Sep 18 2022