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
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;
Programs
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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);
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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.