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

0,2

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.

Links

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

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)).

Example

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 + ...

Prog

(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), ", "))

Crossrefs

Cf. A357158, A357794, A357795.

Sequence in context: A277405 A280572 A217904 * A005330 A003084 A010573

Adjacent sequences: A357793 A357794 A357795 * A357797 A357798 A357799

Keyword

nonn

Author

Paul D. Hanna, Dec 22 2022

Status

approved