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Triangle read by rows: T(m,n) is the number of simple paths for a Racetrack car (using von Neumann neighborhood) with initial velocity zero, going from one corner to the diagonally opposite corner on an m X n grid, 1 <= n <= m.
2

%I #5 Feb 04 2022 14:40:33

%S 1,1,0,1,1,2,2,2,3,8,3,3,7,12,40,5,7,13,26,160,1380,9,13,28,61,918,

%T 12940,211164,14,27,61,161,7260,142453,4997155,205331148

%N Triangle read by rows: T(m,n) is the number of simple paths for a Racetrack car (using von Neumann neighborhood) with initial velocity zero, going from one corner to the diagonally opposite corner on an m X n grid, 1 <= n <= m.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Racetrack_(game)">Racetrack</a>

%e Triangle begins:

%e m\n| 1 2 3 4 5 6 7 8

%e ---+-------------------------------------------

%e 1 | 1

%e 2 | 1 0

%e 3 | 1 1 2

%e 4 | 2 2 3 8

%e 5 | 3 3 7 12 40

%e 6 | 5 7 13 26 160 1380

%e 7 | 9 13 28 61 918 12940 211164

%e 8 | 14 27 61 161 7260 142453 4997155 205331148

%Y Cf. A064297, A291896 (column n=1), A351042, A351106, A351109 (main diagonal).

%K nonn,tabl,more

%O 1,6

%A _Pontus von Brömssen_, Feb 01 2022