A330609 T(n, k) = binomial(n-k-1, k-1)*(n-k)!/k! for n >= 0 and 0 <= k <= floor(n/2). Irregular triangle read by rows.

1, 0, 0, 1, 0, 2, 0, 6, 1, 0, 24, 6, 0, 120, 36, 1, 0, 720, 240, 12, 0, 5040, 1800, 120, 1, 0, 40320, 15120, 1200, 20, 0, 362880, 141120, 12600, 300, 1, 0, 3628800, 1451520, 141120, 4200, 30, 0, 39916800, 16329600, 1693440, 58800, 630, 1

Offset: 0

Author

Peter Luschny, Dec 27 2019

Keywords

• nonn
• tabf

Comments

Also the antidiagonals of the Lah triangle A271703.

Examples

			Triangle begins:
[0] 1
[1] 0
[2] 0, 1
[3] 0, 2
[4] 0, 6,     1
[5] 0, 24,    6
[6] 0, 120,   36,    1
[7] 0, 720,   240,   12
[8] 0, 5040,  1800,  120,  1
[9] 0, 40320, 15120, 1200, 20
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000

Crossrefs

Variants: A180047, A221913. Row sums: A001053.

Cf. A271703.

Programs

Maple

T := (n, k) -> binomial(n-k-1, k-1)*(n-k)!/k!:
seq(seq(T(n, k), k=0..floor(n/2)), n=0..12);
# Alternative:
T := proc(n, k) option remember;
if (n=0 and k=0) or (n=2 and k=1) then 1 elif (k < 1) or (k > ceil(n/2)) then 0
else (n-1)*T(n-1, k) + T(n-2, k-1) fi end: seq(seq(T(n, k), k=0..n/2), n=0..12);

Mathematica

Table[Binomial[n-k-1,k-1] (n-k)!/k!,{n,0,20},{k,0,Floor[n/2]}]//Flatten (* Harvey P. Dale, Oct 19 2021 *)

Formula

T(0,0) = T(2,1) = 1. If k < 1 or k > ceiling(n/2) then T(n,k) = 0. Otherwise:

T(n, k) = (n-1)*T(n-1, k) + T(n-2, k-1)