A286725 Third column of triangle A286724: Lah[2,1](n+2, 2), n >= 0.

1, 18, 288, 4800, 86400, 1693440, 36126720, 836075520, 20901888000, 562028544000, 16186422067200, 497364605337600, 16247243774361600, 562404592189440000, 20567939371499520000, 792551263781781504000, 32098326183162150912000, 1363234794367239585792000

Offset: 0

Author

Wolfdieter Lang, Jun 17 2017

Keywords

• nonn
• easy

Comments

See A286725 for the generalized Lah numbers Lah[2,1].

Crossrefs

Cf. A000165 (m=0), A014479 (m=1), A001620, A051578, A051580, A051582, A286724.

Programs

Mathematica

a[n_] := 2^(n - 1)*(n + 2)! * Binomial[n + 2, 2]; Array[a, 20, 0] (* Amiram Eldar, Dec 11 2022 *)

Formula

E.g.f.: d^2/dx^2 (x^2/(2*(1-2*x)^3)) = (1 + 8*x + 4*x^2)/(1 - 2*x)^5.

a(n) = 2^(n-1)*(n+2)!*binomial(n+2,2), n >= 0.

From Amiram Eldar, Dec 11 2022: (Start)

Sum_{n>=0} 1/a(n) = 32 - 16*sqrt(e) + 8*gamma - 8*Ei(1/2) - 8*log(2), where Ei(x) is the exponential integral and gamma is Euler's constant (A001620).

Sum_{n>=0} (-1)^n/a(n) = 24*gamma - 16/sqrt(e) - 24*Ei(-1/2) - 24*log(2). (End)