A367267 Triangle read by rows. T(n, k) = binomial(n, k) * binomial(n - 1, k - 1).

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

Offset: 0

Author

Peter Luschny, Nov 11 2023

Keywords

• nonn
• tabl

Examples

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

Crossrefs

Cf. A088218 (row sums), A367270 (row reversed).

Programs

Maple

T := (n, k) -> binomial(n, k) * binomial(n - 1, k - 1):
for n from 0 to 6 do seq(T(n, k), k = 0..n) od;
# Or:
T := (n, k) -> if k=0 then k^n elif k=1 then n else (n/k)*binomial(n-1, k-1)^2 fi:
seq(seq(T(n, k), k = 0..n), n = 0..9);

Mathematica

A367267[n_,k_]:=Binomial[n,k]Binomial[n-1,k-1];
Table[A367267[n,k],{n,0,15},{k,0,n}] (* Paolo Xausa, Nov 29 2023 *)

Formula

For k >= 2: T(n, k) = (n / k) * binomial(n-1, k-1)^2.