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%I #16 Dec 12 2015 04:18:17
%S 1,1,3,18,162,1851,24661,365613,5863881,99895425,1785024645,
%T 33156724734,635961987570,12531882072719,252701147866029,
%U 5198011293931270,108793300411597194,2312049376195527621,49804793378882733343,1085910951385068915212,23934948368968158240960
%N 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.
%C 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].
%H Alois P. Heinz, <a href="/A065058/b065058.txt">Table of n, a(n) for n = 0..500</a>
%F a(n) ~ 13 * 3^(3*n+7/2) / (2^11 * Pi * n^4). - _Vaclav Kotesovec_, Sep 10 2014
%e 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].
%t 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)
%Y Cf. A005789.
%K nonn
%O 0,3
%A _Wouter Meeussen_, Nov 06 2001
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