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

0,3

Comments

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

Links

Table of n, a(n) for n=0..44.

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

Example

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.

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, k<n+1, k++, AppendTo[r, (n!)*((n-k+1)^(k-1))*(n+1)/(k!)]]; r
(* columns[k], k[0..oo] *)
k=3; c={}; For[n=k, n<13+k, n++, AppendTo[c, (n!)*((n-k+1)^(k-1))*(n+1)/(k!)]]; c
(* sequence *)
s={}; For[n=0, n<13, n++, For[k=0, k<n+1, k++, AppendTo[s, (n!)*((n-k+1)^(k-1))*(n+1)/(k!)]]]; s
(* Detlef Meya, Jul 31 2023 *)

Crossrefs

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

Sequence in context: A100641 A028421 A263003 * A308159 A081745 A240578

Adjacent sequences: A344466 A344467 A344468 * A344470 A344471 A344472

Keyword

nonn,tabl

Author

Peter Luschny, May 20 2021

Status

approved