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

0,6

Comments

Also the antidiagonals of the Lah triangle A271703.

Links

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

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)

Example

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

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 *)

Crossrefs

Variants: A180047, A221913. Row sums: A001053.

Cf. A271703.

Sequence in context: A361522 A137437 A183189 * A180047 A180397 A347133

Adjacent sequences: A330606 A330607 A330608 * A330610 A330611 A330612

Keyword

nonn,tabf

Author

Peter Luschny, Dec 27 2019

Status

approved