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

1,1

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.

Links

Table of n, a(n) for n=1..14.

Formula

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

Example

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.

Sequence in context: A050984 A169686 A184889 * A300147 A087216 A239925

Adjacent sequences: A358478 A358479 A358480 * A358482 A358483 A358484

Keyword

nonn

Author

Janaka Rodrigo, Nov 18 2022

Status

approved