A344469 Triangle read by rows: T(n, k) (0 <= k <= n) = [x^k] x^n * n! * [t^n] x*(1 + t)/(x*exp(-t) - t).

1, 1, 2, 2, 6, 3, 6, 24, 24, 4, 24, 120, 180, 80, 5, 120, 720, 1440, 1080, 240, 6, 720, 5040, 12600, 13440, 5670, 672, 7, 5040, 40320, 120960, 168000, 107520, 27216, 1792, 8, 40320, 362880, 1270080, 2177280, 1890000, 774144, 122472, 4608, 9

Offset: 0

Author

Peter Luschny, May 20 2021

Keywords

• nonn
• tabl

Comments

Related to the Lambert W-function, see Cohen, Corollary 2.4.

Examples

			Triangle starts:
[0] 1;
[1] 1,     2;
[2] 2,     6,      3;
[3] 6,     24,     24,      4;
[4] 24,    120,    180,     80,      5;
[5] 120,   720,    1440,    1080,    240,     6;
[6] 720,   5040,   12600,   13440,   5670,    672,    7;
[7] 5040,  40320,  120960,  168000,  107520,  27216,  1792,   8;
[8] 40320, 362880, 1270080, 2177280, 1890000, 774144, 122472, 4608, 9.
		

Links

• Henri Cohen, Lambert W-Function Branch Identities, arXiv:2012.11698v2 [math.CV], 2020-2021.

Crossrefs

Cf. A305990 (row sums), A009306 (alternating row sums).

Programs

Maple

gf := x*(1+t)/(x*exp(-t)-t): ser := series(gf,t,12):
seq(seq(coeff(expand(x^n*n!*coeff(ser,t,n)),x,k),k=0..n),n=0..8);

Mathematica

(* rows[n], n[0..oo] *)
n=12;r={};For[k=0,kDetlef Meya, Jul 31 2023 *)