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A357226 Coefficients in the power series A(x) such that: x*A(x)^6 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) * A(x)^n.

%I A357226 #5 Sep 19 2022 11:09:17
%S A357226 1,1,7,73,861,11112,151822,2159143,31627140,473909468,7230035454,
%T A357226 111924733904,1753728878625,27759947012294,443247756591472,
%U A357226 7130680715081049,115466397372003479,1880525144522628300,30783524695736369568,506215648672559259036,8358521379108937920413
%N A357226 Coefficients in the power series A(x) such that: x*A(x)^6 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) * A(x)^n.
%F A357226 G.f. A(x) satisfies:
%F A357226 (1) x*A(x)^6 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) * A(x)^n.
%F A357226 (2) -x*A(x)^7 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) / A(x)^n.
%F A357226 (3) x*A(x)^6 = 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 A357226 (4) -x*A(x)^7 = 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 A357226 G.f.: A(x) = 1 + x + 7*x^2 + 73*x^3 + 861*x^4 + 11112*x^5 + 151822*x^6 + 2159143*x^7 + 31627140*x^8 + 473909468*x^9 + 7230035454*x^10 + ...
%e A357226 such that
%e A357226 x*A(x)^6 = ... + 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 A357226 (PARI) {a(n,p=6) = my(A=[1]); for(i=1, n, A=concat(A, 0);
%o A357226 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 A357226 for(n=0, 30, print1(a(n), ", "))
%Y A357226 Cf. A355357, A357221, A357222, A357223, A357224, A357225.
%K A357226 nonn
%O A357226 0,3
%A A357226 _Paul D. Hanna_, Sep 18 2022