A357225 - OEIS

%I #5 Sep 19 2022 11:09:38

%S 1,1,6,54,542,5950,69089,834807,10387628,132206325,1713016233,

%T 22520857313,299667203315,4028078782339,54615552455056,

%U 746073353306341,10258385111897258,141862903772876529,1971827463536643265,27532294076219156008,386001188585539328720

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

%F (1) x*A(x)^5 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) * A(x)^n.

%F (2) -x*A(x)^6 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) / A(x)^n.

%F (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 (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 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 such that

%e 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 (PARI) {a(n,p=5) = 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, A357222, A357223, A357224, A357226.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Sep 18 2022