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
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;
Programs
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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);
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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.