A360230 a(n) = coefficient of x^n/n! in Sum_{n>=0} (1 + n*x + x^2)^n * x^n/n!.

1, 1, 3, 19, 109, 921, 8911, 100003, 1307769, 18748369, 307713691, 5379610611, 106277271013, 2194176659689, 50689643777319, 1207518763542211, 31940171681228401, 862606920178886433, 25708097594461923379, 776354747057987797459, 25741373454075987900381

Offset

0,3

Links

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

Formula

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! may be defined as follows.

(1) A(x) = Sum_{n>=0} (1 + n*x + x^2)^n * x^n/n!.

(2) A(x) = ( LambertW(-x^2)/(-x^2) )^(x + 1/x) / (1 + LambertW(-x^2)).

(3) A(x) = exp( -(x + 1/x) * LambertW(-x^2) ) / (1 + LambertW(-x^2)).

a(n) ~ (exp(exp(1/2) + exp(-1/2)) + (-1)^n*exp(- exp(1/2) - exp(-1/2))) * n^n / (sqrt(2) * exp(n/2)). - Vaclav Kotesovec, Mar 14 2023

Example

E.g.f.: A(x) = 1 + x + 3*x^2/2! + 19*x^3/3! + 109*x^4/4! + 921*x^5/5! + 8911*x^6/6! + 100003*x^7/7! + 1307769*x^8/8! + 18748369*x^9/9! + ...
such that
A(x) = 1 + (1 + x + x^2)*x + (1 + 2*x + x^2)^2*x^2/2! + (1 + 3*x + x^2)^3*x^3/3! + (1 + 4*x + x^2)^4*x^4/4! + (1 + 5*x + x^2)^5*x^5/5! + ... + (1 + n*x + x^2)^n * x^n/n! + ...
also
A(x) = W(x^2)^(x + 1/x) / (1 - x^2*W(x^2)), where W(x) = LambertW(-x)/(-x).

Prog

(PARI) {a(n) = n! * polcoeff( sum(m=0, n, (1 + m*x + x^2)^m * x^m/m! +x*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = n! * polcoeff( ( lambertw(-x^2 + O(x^(n+6)))/(-x^2) )^(x + 1/x) / (1 + lambertw(-x^2 + O(x^(n+6)))), n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A000169.

Sequence in context: A103005 A162354 A132959 * A253811 A037154 A037774

Adjacent sequences: A360227 A360228 A360229 * A360231 A360232 A360233

Keyword

nonn

Author

Paul D. Hanna, Feb 19 2023

Status

approved