A367270 Triangle read by rows. T(n, k) = binomial(n, k)*binomial(n - 1, n - k - 1).

1, 1, 0, 1, 2, 0, 1, 6, 3, 0, 1, 12, 18, 4, 0, 1, 20, 60, 40, 5, 0, 1, 30, 150, 200, 75, 6, 0, 1, 42, 315, 700, 525, 126, 7, 0, 1, 56, 588, 1960, 2450, 1176, 196, 8, 0, 1, 72, 1008, 4704, 8820, 7056, 2352, 288, 9, 0, 1, 90, 1620, 10080, 26460, 31752, 17640, 4320, 405, 10, 0

Offset: 0

Author

Peter Luschny, Nov 11 2023

Keywords

• nonn
• tabl

Examples

			Triangle T(n, k) begins:
  [0] 1;
  [1] 1,  0;
  [2] 1,  2,    0;
  [3] 1,  6,    3,    0;
  [4] 1, 12,   18,    4,    0;
  [5] 1, 20,   60,   40,    5,    0;
  [6] 1, 30,  150,  200,   75,    6,    0;
  [7] 1, 42,  315,  700,  525,  126,    7,   0;
  [8] 1, 56,  588, 1960, 2450, 1176,  196,   8, 0;
  [9] 1, 72, 1008, 4704, 8820, 7056, 2352, 288, 9, 0;
		

Crossrefs

Cf. A088218 (row sums), A367267 (row reversed).

Programs

Maple

T := (n, k) -> binomial(n, k) * binomial(n - 1, n - k - 1):
# Or:
T := (n, k) -> if k=0 then 1 elif k=n then 0 else ((n-k)/n)*binomial(n, k)^2 fi:
seq(seq(T(n, k), k = 0..n), n = 0..9);

Mathematica

A367270[n_,k_]:=Binomial[n,k]Binomial[n-1,n-k-1];
Table[A367270[n,k],{n,0,15},{k,0,n}] (* Paolo Xausa, Nov 29 2023 *)

Formula

For 0< k < n: T(n, k) = ((n - k) / n)*binomial(n, k)^2.