OFFSET
0,3
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].
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..500
FORMULA
a(n) ~ 13 * 3^(3*n+7/2) / (2^11 * Pi * n^4). - Vaclav Kotesovec, Sep 10 2014
EXAMPLE
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].
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)
CROSSREFS
KEYWORD
nonn
AUTHOR
Wouter Meeussen, Nov 06 2001
STATUS
approved