cp's OEIS Frontend

A357225 Coefficients in the power series A(x) such that: x*A(x)^5 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) * A(x)^n.

%I A357225 #5 Sep 19 2022 11:09:38
%S A357225 1,1,6,54,542,5950,69089,834807,10387628,132206325,1713016233,
%T A357225 22520857313,299667203315,4028078782339,54615552455056,
%U A357225 746073353306341,10258385111897258,141862903772876529,1971827463536643265,27532294076219156008,386001188585539328720
%N A357225 Coefficients in the power series A(x) such that: x*A(x)^5 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) * A(x)^n.
%F A357225 G.f. A(x) satisfies:
%F A357225 (1) x*A(x)^5 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) * A(x)^n.
%F A357225 (2) -x*A(x)^6 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) / A(x)^n.
%F A357225 (3) 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.
%F A357225 (4) -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.
%e A357225 G.f.: A(x) = 1 + x + 6*x^2 + 54*x^3 + 542*x^4 + 5950*x^5 + 69089*x^6 + 834807*x^7 + 10387628*x^8 + 132206325*x^9 + 1713016233*x^10 + ...
%e A357225 such that
%e A357225 x*A(x)^5 = ... + 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 A357225 (PARI) {a(n,p=5) = my(A=[1]); for(i=1, n, A=concat(A, 0);
%o A357225 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 A357225 for(n=0, 30, print1(a(n), ", "))
%Y A357225 Cf. A355357, A357221, A357222, A357223, A357224, A357226.
%K A357225 nonn
%O A357225 0,3
%A A357225 _Paul D. Hanna_, Sep 18 2022