A065058 Number of paths to T(n,n,n) with T(i,j,k)= 0 if j>i or k>j and T(i,j,k) = T(i-1,j,k) + T(i,j-1,k) + T(i,j,k-1) and T(i,j,0) = 1.

1, 1, 3, 18, 162, 1851, 24661, 365613, 5863881, 99895425, 1785024645, 33156724734, 635961987570, 12531882072719, 252701147866029, 5198011293931270, 108793300411597194, 2312049376195527621, 49804793378882733343, 1085910951385068915212, 23934948368968158240960

Offset: 0

Author

Wouter Meeussen, Nov 06 2001

Keywords

• nonn

Comments

Similar to the "3-dimensional Catalan numbers" of A005789, but with paths starting from anywhere on z=0, instead of only from [0,0,0].

Examples

			a(3) = 18 because [3,3,3] can be reached from [x,y,0] in the following ways (along nondecreasing paths): 5 [1,1,0] + 5 [2,1,0] + 3 [2,2,0] + 2 [3,1,0] + 2 [3,2,0] + [3,3,0].
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A005789.

Programs

Mathematica

T[0, 0, 0] := 1; T[x_, y_, z_] := 0 /; (x< y || y< z); T[u_, v_, 0] := 1; T[, 0, 0] := 1 T[x, y_, z_] := (T[x, y, z]= T[x-1, y, z]+T[x, y-1, z] +T[x, y, z-1]) /; (y<=x ||z<=y)

Formula

a(n) ~ 13 * 3^(3*n+7/2) / (2^11 * Pi * n^4). - Vaclav Kotesovec, Sep 10 2014