|
%I #7 Sep 19 2022 11:08:03
%S 1,1,9,96,1150,14981,206426,2959249,43683374,659531482,10137150414,
%T 158089344305,2495255246353,39785814006395,639880150931025,
%U 10368454503796731,169106511176489353,2773945868018478593,45734618620228469488,757469141505480597690
%N Coefficients in the power series A(x) such that: x*A(x)^5 = 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)^5 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.
%F (2) -x*A(x)^6 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) / A(x)^n.
%F (3) 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.
%F (4) -x*A(x)^6 = 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 + 9*x^2 + 96*x^3 + 1150*x^4 + 14981*x^5 + 206426*x^6 + 2959249*x^7 + 43683374*x^8 + 659531482*x^9 + 10137150414*x^10 + ...
%e where
%e x*A(x)^5 = ... - 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)^5 - 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, A357208.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Sep 18 2022
| 