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A365511 Number of ways to travel from (0,0,0) to (2*n,2*n,2*n) with n positive integer steps in each direction, changing directions at each step.

%I A365511 #9 Sep 30 2023 21:47:08
%S A365511 1,6,810,174000,46819500,14378702688,4817350825056,1716615248325120,
%T A365511 640480159385995500,247630745402467284000,98500241916182188189536,
%U A365511 40099260132768751505660160,16642069286080355216946537600,7020218653006514588616480000000,3002947242700351209440983200000000
%N A365511 Number of ways to travel from (0,0,0) to (2*n,2*n,2*n) with n positive integer steps in each direction, changing directions at each step.
%C A365511 Rotations or reflections are counted as different paths. For example, when n=1 then there are six paths from (0,0,0) to (2,2,2) using one step in each direction; these six paths would correspond to the six permutations of x,y,z, which are xyz, xzy, yxz, yzx, zxy, zyx. If we discount rotations then there would be just two paths: xyz and xzy. If we discount reflections, there would be just one path: xyz.
%F A365511 a(n) = A088218(n)^3 * A110706(n).
%e A365511 For n=2, we consider all possible paths from (0,0,0) to (4,4,4) involving two steps in each coordinate direction. We can begin this count by considering all the ways to arrange two x's, two y's, and two z's without consecutive terms; there are 30 such ways because A110706(2) = 30. Then, for the two x's which represent the two steps in the x-direction, they need to add up to 4 and there are three such ways (1+3, 2+2, and 3+1). Likewise there are three ways for the y's and likewise three ways for the z's. Hence, in total, we have 3*3*3*30 = 810 ways to move from (0,0,0) to (4,4,4) with two steps in each direction with no two consecutive steps in same direction.
%Y A365511 Cf. A088218, A110706.
%K A365511 nonn
%O A365511 0,2
%A A365511 _Greg Dresden_ and Snezhana Tuneska, Sep 07 2023