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].

1, 1, 1, 1, 3, 5, 1, 6, 21, 42, 1, 10, 56, 210, 462, 1, 15, 120, 660, 2574, 6006, 1, 21, 225, 1650, 9009, 36036, 87516, 1, 28, 385, 3575, 25025, 136136, 554268, 1385670, 1, 36, 616, 7007, 60060, 408408, 2217072, 9145422, 23371634, 1, 45, 936, 12740, 129948

Offset

0,5

Comments

T(n,m)= number of standard tableaux of shape (n,m,m) (0<m<=n). - Emeric Deutsch, May 14 2004

Links

Table of n, a(n) for n=0..49.

Formula

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}]

T(n, m)=(n+2m)!(n-m+1)(n-m+2)/[m!(m+1)!(n+2)! ] (0<=m<=n). - Emeric Deutsch, May 14 2004

Example

1;
1,1;
1,3,5;
1,6,21,42;
1,10,56,210,462;
1,15,120,660,2574,6006;
...
T(2,1)=3 because in the first row of the diagram (2,1,1) we can have 12 or 13 or 14.

Maple

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);

Crossrefs

Last number in each row is A005789

Cf. A005789, A065058, A065078.

Sequence in context: A073365 A316152 A302204 * A118788 A196020 A346775

Adjacent sequences: A065074 A065075 A065076 * A065078 A065079 A065080

Keyword

nonn,tabl

Author

Wouter Meeussen, Nov 09 2001

Status

approved