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Triangle read by rows: T(n,k) is the number of standard tableaux of shape (n,n,k) (0<=k<=n).
0

%I #7 Apr 09 2013 11:32:09

%S 1,1,1,2,5,5,5,21,42,42,14,84,252,462,462,42,330,1320,3432,6006,6006,

%T 132,1287,6435,21450,51480,87516,87516,429,5005,30030,121550,364650,

%U 831402,1385670,1385670,1430,19448,136136,646646,2309450,6466460

%N Triangle read by rows: T(n,k) is the number of standard tableaux of shape (n,n,k) (0<=k<=n).

%C Column k=0 is the sequence of Catalan numbers (A000108).

%F T(n, k)=(2n+k)!(n-k+2)(n-k+1)/[k!(n+2)!(n+1)! ] (0<=k<=n).

%e 1;

%e 1,1;

%e 2,5,5;

%e 5,21,42,42;

%e 14,84,252,462,462;

%e 42,330,1320,3432,6006,6006;

%p T:=proc(n,k) if k>n then 0 else (2*n+k)!*(n-k+2)*(n-k+1)/k!/(n+2)!/(n+1)! fi end:seq(seq(T(n,k),k=0..n),n=0..9);

%Y Cf. A000108.

%K nonn,tabl

%O 0,4

%A _Emeric Deutsch_, May 30 2004

%E Keyword tabl added by _Michel Marcus_, Apr 09 2013