A358481 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.

30, 6218, 2658432, 1054788750, 552306591900, 269380692717696, 155175092086118400, 83798883891736779150, 50885239237727996887500, 29198209396114625497699068, 18332853214682572877389897728, 10951674446687597386319569942656, 7036938452279110885561897815723264, 4325988198220149508865311059521280000

Offset: 1

Author

Janaka Rodrigo, Nov 18 2022

Keywords

• nonn

Comments

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.

Examples

			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.
Number of pairs which have the middle point of their paths as a common point are A357760(2)=1782.
Therefore number of pairs without having middle point on their paths as a common point are 8100-1782=6218
		

Crossrefs

Cf. A268553, A357760.

Formula

a(n) = A268553(n) - A357760(n).