A357796 a(n) = coefficient of x^n in the power series A(x) such that: 1 = Sum_{n=-oo..+oo} n*(n+1)*(n+2)*(n+3)/4! * x^n * (1 - x^(n+3))^n * A(x)^(n+3).

1, 5, 40, 635, 12095, 248245, 5381435, 121355095, 2817706420, 66909209195, 1617401484401, 39668321722180, 984661725380420, 24690230217076810, 624476169158179615, 15912858189842638180, 408139640637624168780, 10528308534373198776840, 272970775748658547320275

Offset: 0

Author

Paul D. Hanna, Dec 22 2022

Keywords

• nonn

Comments

Related identity: 0 = Sum_{n=-oo..+oo} n*(n+1)*(n+2)*(n+3)/4! * x^(4*n) * (y - x^n)^(n-1), which holds formally for all y.

Examples

			G.f.: A(x) = 1 + 5*x + 40*x^2 + 635*x^3 + 12095*x^4 + 248245*x^5 + 5381435*x^6 + 121355095*x^7 + 2817706420*x^8 + 66909209195*x^9 + 1617401484401*x^10 + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A357158, A357794, A357795.

Programs

PARI

{a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( sum(n=-#A-2, #A+2, n*(n+1)*(n+2)*(n+3)/4! * x^n * if(n==-3,0, (1 - x^(n+3) +x*O(x^#A) )^n) * Ser(A)^(n+3) ), #A-1) ); H=A; A[n+1]}
for(n=0, 30, print1(a(n), ", "))

PARI

{a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( sum(n=-#A-2, #A+2, (-1)^n * n*(n-1)*(n-2)*(n-3)/4! * x^(n*(n-4)) * if(n==3,0, 1/(1 - x^(n-3) +x*O(x^#A) )^n) / Ser(A)^(n-3) ), #A-1) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

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

(1) 1 = Sum_{n=-oo..+oo} n*(n+1)*(n+2)*(n+3)/4! * x^n * (1 - x^(n+3))^n * A(x)^(n+3).

(2) 1 = Sum_{n=-oo..+oo} (-1)^n * n*(n-1)*(n-2)*(n-3)/4! * x^(n*(n-4)) / ((1 - x^(n-3))^n * A(x)^(n-3)).