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A357224 Coefficients in the power series A(x) such that: x*A(x)^4 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) * A(x)^n.

%I A357224 #5 Sep 19 2022 11:16:24
%S A357224 1,1,5,38,315,2855,27325,272030,2788042,29221793,311767823,3374650902,
%T A357224 36968040004,409076635878,4565873250981,51342245169913,
%U A357224 581093383193700,6614534942714496,75675364150733073,869713202188274489,10036085000519702155,116238137830534589525
%N A357224 Coefficients in the power series A(x) such that: x*A(x)^4 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) * A(x)^n.
%F A357224 G.f. A(x) satisfies:
%F A357224 (1) x*A(x)^4 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) * A(x)^n.
%F A357224 (2) -x*A(x)^5 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) / A(x)^n.
%F A357224 (3) x*A(x)^4 = Product_{n>=1} (1 - x^(2*n)*A(x)) * (1 - x^(2*n-2)/A(x)) * (1 - x^(2*n)), due to the Jacobi triple product identity.
%F A357224 (4) -x*A(x)^5 = Product_{n>=1} (1 - x^(2*n)/A(x)) * (1 - x^(2*n-2)*A(x)) * (1 - x^(2*n)), due to the Jacobi triple product identity.
%e A357224 G.f.: A(x) = 1 + x + 5*x^2 + 38*x^3 + 315*x^4 + 2855*x^5 + 27325*x^6 + 272030*x^7 + 2788042*x^8 + 29221793*x^9 + 311767823*x^10 + ...
%e A357224 such that
%e A357224 x*A(x)^4 = ... + x^12/A(x)^4 - x^6/A(x)^3 + x^2/A(x)^2 - 1/A(x) + 1 - x^2*A(x) + x^6*A(x)^2 - x^12*A(x)^3 + x^20*A(x)^4 + ... + (-1)^n * x^(n*(n+1)) * A(x)^n + ...
%o A357224 (PARI) {a(n,p=4) = my(A=[1]); for(i=1, n, A=concat(A, 0);
%o A357224 A[#A] = polcoeff( x*Ser(A)^p - sum(m=-ceil(sqrt(n)), ceil(sqrt(n)), (-1)^m*x^(m*(m+1))*Ser(A)^m ), #A-1)); A[n+1]}
%o A357224 for(n=0, 30, print1(a(n), ", "))
%Y A357224 Cf. A355357, A357221, A357222, A357223, A357225, A357226.
%K A357224 nonn
%O A357224 0,3
%A A357224 _Paul D. Hanna_, Sep 18 2022