A357207 - OEIS

%I #7 Sep 19 2022 11:09:01

%S 1,1,7,55,469,4307,41678,418872,4330275,45754091,491916135,5364166402,

%T 59186372395,659556170091,7412556531714,83921355689635,

%U 956228695216241,10957322339242547,126189988012692329,1459793848341094130,16955390069787782159,197653935181097885580

%N Coefficients in the power series A(x) such that: x*A(x)^3 = 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)^3 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.

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

%F (3) x*A(x)^3 = 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)^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.

%e G.f.: A(x) = 1 + x + 7*x^2 + 55*x^3 + 469*x^4 + 4307*x^5 + 41678*x^6 + 418872*x^7 + 4330275*x^8 + 45754091*x^9 + 491916135*x^10 + ...

%e where

%e x*A(x)^3 = ... - 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)^3 - 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, A357208, A357209.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Sep 18 2022