%I #8 Dec 21 2022 12:03:07
%S 30,6218,2658432,1054788750,552306591900,269380692717696,
%T 155175092086118400,83798883891736779150,50885239237727996887500,
%U 29198209396114625497699068,18332853214682572877389897728,10951674446687597386319569942656,7036938452279110885561897815723264,4325988198220149508865311059521280000
%N a(n) is the number of different pairs of shortest grid paths joining two opposite corners in opposite order in an n X n X n grid without having middle point on their paths as a common point.
%C Alternatively a(n) is the number of ways two ants can interchange their positions starting simultaneously from two opposite corners and moving along shortest grid paths at same speed in an n X n X n grid without meeting other one.
%F a(n) = A268553(n)  A357760(n).
%e When n=2 number of ways to move between two opposite corners are given by 6!/(2!*2!*2!) and number of such pairs are given by (6!/(2!*2!*2!))^2. This total number of pairs are given by A268553(2)=8100.
%e Number of pairs which have the middle point of their paths as a common point are A357760(2)=1782.
%e Therefore number of pairs without having middle point on their paths as a common point are 81001782=6218
%Y Cf. A268553, A357760.
%K nonn
%O 1,1
%A _Janaka Rodrigo_, Nov 18 2022
