A357794 a(n) = coefficient of x^n in the power series A(x) such that: 1 = Sum_{n=-oo..+oo} n*(n+1)/2 * x^n * (1 - x^(n+1))^n * A(x)^(n+1).

1, 3, 15, 114, 1086, 10824, 114382, 1252002, 14083275, 161810358, 1890774909, 22401092826, 268465408738, 3248818848876, 39643793276526, 487251937616006, 6026537732208078, 74954027622814455, 936840765257368687, 11761260253206563461, 148240496011414115676

Offset

0,2

Comments

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

Links

Paul D. Hanna, Table of n, a(n) for n = 0..300

Formula

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

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

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

Example

G.f.: A(x) = 1 + 3*x + 15*x^2 + 114*x^3 + 1086*x^4 + 10824*x^5 + 114382*x^6 + 1252002*x^7 + 14083275*x^8 + 161810358*x^9 + 1890774909*x^10 + ...

Prog

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

Crossrefs

Cf. A357158, A357795, A357796.

Sequence in context: A343707 A059849 A123853 * A335531 A166885 A346224

Adjacent sequences: A357791 A357792 A357793 * A357795 A357796 A357797

Keyword

nonn

Author

Paul D. Hanna, Dec 22 2022

Status

approved