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A065077 Triangle read by rows: T(n,m) = C[n,m,m] where C[i,j,k] is the 3-dimensional Catalan pyramid defined by C[0,0,0]=1 and C[i,j,k]=0 if j>i or k>j and C[i,j,k]=C[i-1,j,k]+C[i,j-1,k]+C[i,j,k-1].

%I A065077 #8 Aug 10 2015 07:33:48
%S A065077 1,1,1,1,3,5,1,6,21,42,1,10,56,210,462,1,15,120,660,2574,6006,1,21,
%T A065077 225,1650,9009,36036,87516,1,28,385,3575,25025,136136,554268,1385670,
%U A065077 1,36,616,7007,60060,408408,2217072,9145422,23371634,1,45,936,12740,129948
%N A065077 Triangle read by rows: T(n,m) = C[n,m,m] where C[i,j,k] is the 3-dimensional Catalan pyramid defined by C[0,0,0]=1 and C[i,j,k]=0 if j>i or k>j and C[i,j,k]=C[i-1,j,k]+C[i,j-1,k]+C[i,j,k-1].
%C A065077 T(n,m)= number of standard tableaux of shape (n,m,m) (0<m<=n). - _Emeric Deutsch_, May 14 2004
%F A065077 C[0, 0, 0] := 1; C[x_, y_, z_] := 0 /; (x< y || y< z); C[u_, v_, 0] := (u+v)!/(u+1)!/(v)!(u-v+1); C[_, 0, 0] := 1; C[x_, y_, z_] := (C[x, y, z]= C[x-1, y, z]+C[x, y-1, z] +C[x, y, z-1]) /; (y<=x ||z<=y); Table[C[x, y, y], {x, 0, 10}, {y, 0, x}]
%F A065077 T(n, m)=(n+2m)!(n-m+1)(n-m+2)/[m!(m+1)!(n+2)! ] (0<=m<=n). - _Emeric Deutsch_, May 14 2004
%e A065077 1;
%e A065077 1,1;
%e A065077 1,3,5;
%e A065077 1,6,21,42;
%e A065077 1,10,56,210,462;
%e A065077 1,15,120,660,2574,6006;
%e A065077 ...
%e A065077 T(2,1)=3 because in the first row of the diagram (2,1,1) we can have 12 or 13 or 14.
%p A065077 a:=proc(n,m) if m<=n then (n+2*m)!*(n-m+1)*(n-m+2)/m!/(m+1)!/(n+2)! else 0 fi end: seq(seq(a(n,m),m=0..n),n=0..9);
%Y A065077 Last number in each row is A005789
%Y A065077 Cf. A005789, A065058, A065078.
%K A065077 nonn,tabl
%O A065077 0,5
%A A065077 _Wouter Meeussen_, Nov 09 2001