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Coefficients in the power series A(x) such that: x*A(x)^2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) * A(x)^n.
6

%I #5 Sep 19 2022 11:16:00

%S 1,1,3,15,73,391,2180,12620,75056,456004,2817879,17656517,111919061,

%T 716379379,4623944175,30062540989,196692237527,1294112710358,

%U 8556766562091,56829292404053,378936456243142,2535866861527016,17025875430611442,114654511539186113

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

%F G.f. A(x) satisfies:

%F (1) x*A(x)^2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) * A(x)^n.

%F (2) -x*A(x)^3 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) / A(x)^n.

%F (3) x*A(x)^2 = 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 (4) -x*A(x)^3 = 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 G.f.: A(x) = 1 + x + 3*x^2 + 15*x^3 + 73*x^4 + 391*x^5 + 2180*x^6 + 12620*x^7 + 75056*x^8 + 456004*x^9 + 2817879*x^10 + ...

%e such that

%e x*A(x)^2 = ... + 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 (PARI) {a(n,p=2) = my(A=[1]); for(i=1, n, A=concat(A, 0);

%o 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 for(n=0, 30, print1(a(n), ", "))

%Y Cf. A355357, A357221, A357223, A357224, A357225, A357226.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Sep 18 2022