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

%I A367270 #9 Nov 29 2023 11:04:11
%S A367270 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,
%T A367270 42,315,700,525,126,7,0,1,56,588,1960,2450,1176,196,8,0,1,72,1008,
%U A367270 4704,8820,7056,2352,288,9,0,1,90,1620,10080,26460,31752,17640,4320,405,10,0
%N A367270 Triangle read by rows. T(n, k) = binomial(n, k)*binomial(n - 1, n - k - 1).
%F A367270 For 0< k < n: T(n, k) = ((n - k) / n)*binomial(n, k)^2.
%e A367270 Triangle T(n, k) begins:
%e A367270   [0] 1;
%e A367270   [1] 1,  0;
%e A367270   [2] 1,  2,    0;
%e A367270   [3] 1,  6,    3,    0;
%e A367270   [4] 1, 12,   18,    4,    0;
%e A367270   [5] 1, 20,   60,   40,    5,    0;
%e A367270   [6] 1, 30,  150,  200,   75,    6,    0;
%e A367270   [7] 1, 42,  315,  700,  525,  126,    7,   0;
%e A367270   [8] 1, 56,  588, 1960, 2450, 1176,  196,   8, 0;
%e A367270   [9] 1, 72, 1008, 4704, 8820, 7056, 2352, 288, 9, 0;
%p A367270 T := (n, k) -> binomial(n, k) * binomial(n - 1, n - k - 1):
%p A367270 # Or:
%p A367270 T := (n, k) -> if k=0 then 1 elif k=n then 0 else ((n-k)/n)*binomial(n, k)^2 fi:
%p A367270 seq(seq(T(n, k), k = 0..n), n = 0..9);
%t A367270 A367270[n_,k_]:=Binomial[n,k]Binomial[n-1,n-k-1];
%t A367270 Table[A367270[n,k],{n,0,15},{k,0,n}] (* _Paolo Xausa_, Nov 29 2023 *)
%Y A367270 Cf. A088218 (row sums), A367267 (row reversed).
%K A367270 nonn,tabl
%O A367270 0,5
%A A367270 _Peter Luschny_, Nov 11 2023