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

%I A367267 #9 Nov 29 2023 11:03:05
%S A367267 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,
%T A367267 7,126,525,700,315,42,1,0,8,196,1176,2450,1960,588,56,1,0,9,288,2352,
%U A367267 7056,8820,4704,1008,72,1,0,10,405,4320,17640,31752,26460,10080,1620,90,1
%N A367267 Triangle read by rows. T(n, k) = binomial(n, k) * binomial(n - 1, k - 1).
%F A367267 For k >= 2: T(n, k) = (n / k) * binomial(n-1, k-1)^2.
%e A367267 Triangle T(n, k) starts:
%e A367267   [0] 1;
%e A367267   [1] 0, 1;
%e A367267   [2] 0, 2,   1;
%e A367267   [3] 0, 3,   6,    1;
%e A367267   [4] 0, 4,  18,   12,    1;
%e A367267   [5] 0, 5,  40,   60,   20,    1;
%e A367267   [6] 0, 6,  75,  200,  150,   30,    1;
%e A367267   [7] 0, 7, 126,  525,  700,  315,   42,    1;
%e A367267   [8] 0, 8, 196, 1176, 2450, 1960,  588,   56,  1;
%e A367267   [9] 0, 9, 288, 2352, 7056, 8820, 4704, 1008, 72, 1;
%p A367267 T := (n, k) -> binomial(n, k) * binomial(n - 1, k - 1):
%p A367267 for n from 0 to 6 do seq(T(n, k), k = 0..n) od;
%p A367267 # Or:
%p A367267 T := (n, k) -> if k=0 then k^n elif k=1 then n else (n/k)*binomial(n-1, k-1)^2 fi:
%p A367267 seq(seq(T(n, k), k = 0..n), n = 0..9);
%t A367267 A367267[n_,k_]:=Binomial[n,k]Binomial[n-1,k-1];
%t A367267 Table[A367267[n,k],{n,0,15},{k,0,n}] (* _Paolo Xausa_, Nov 29 2023 *)
%Y A367267 Cf. A088218 (row sums), A367270 (row reversed).
%K A367267 nonn,tabl
%O A367267 0,5
%A A367267 _Peter Luschny_, Nov 11 2023