A357208 - OEIS

%I #7 Sep 19 2022 11:08:34

%S 1,1,8,74,758,8412,98605,1201739,15075377,193374064,2524704727,

%T 33440460233,448246477551,6069174992443,82884604316537,

%U 1140361539606239,15791577929661603,219930850717175458,3078540089119391233,43287917046150591163,611156850554916771425

%N Coefficients in the power series A(x) such that: x*A(x)^4 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.

%F G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following relations.

%F (1) x*A(x)^4 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.

%F (2) -x*A(x)^5 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) / A(x)^n.

%F (3) x*A(x)^4 = Product_{n>=1} (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)) * (1 - x^n), due to the Jacobi triple product identity.

%F (4) -x*A(x)^5 = Product_{n>=1} (1 - x^n/A(x)) * (1 - x^(n-1)*A(x)) * (1 - x^n), due to the Jacobi triple product identity.

%e G.f.: A(x) = 1 + x + 8*x^2 + 74*x^3 + 758*x^4 + 8412*x^5 + 98605*x^6 + 1201739*x^7 + 15075377*x^8 + 193374064*x^9 + 2524704727*x^10 + ...

%e where

%e x*A(x)^4 = ... - x^10/A(x)^5 + x^6/A(x)^4 - x^3/A(x)^3 + x/A(x)^2 - 1/A(x) + 1 - x*A(x) + x^3*A(x)^2 - x^6*A(x)^3 + x^10*A(x)^4 -+ ... + (-1)^n * x^(n*(n+1)/2) * A(x)^n + ...

%o (PARI) {a(n) = my(A=[1, 1], t); for(i=1, n, A=concat(A, 0); t = ceil(sqrt(2*n+9));

%o A[#A] = polcoeff( x*Ser(A)^4 - sum(m=-t, t, (-1)^m*x^(m*(m+1)/2)*Ser(A)^m ), #A-1)); A[n+1]}

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

%Y Cf. A355361, A357206, A357207, A357209.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Sep 18 2022