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T(n,k) is the number of n X k nonnegative integer arrays with new values 0 upwards introduced in row major order and no element equal to more than two of its immediate leftward or upward or right-upward antidiagonal neighbors.
12

%I #8 Oct 14 2024 15:37:28

%S 1,2,2,5,15,5,15,198,203,15,52,4041,20746,4140,52,203,113458,4132120,

%T 4150760,115975,203,877,4132120,1358524513,10318694804,1366230232,

%U 4213597,877,4140,187612143,671329819215,50996571454200,51074630353994

%N T(n,k) is the number of n X k nonnegative integer arrays with new values 0 upwards introduced in row major order and no element equal to more than two of its immediate leftward or upward or right-upward antidiagonal neighbors.

%H R. H. Hardin, <a href="/A208466/b208466.txt">Table of n, a(n) for n = 1..58</a>

%e Table starts:

%e ....1...........2..................5.....................15

%e ....2..........15................198...................4041

%e ....5.........203..............20746................4132120

%e ...15........4140............4150760............10318694804

%e ...52......115975.........1366230232.........51074630353994

%e ..203.....4213597.......675203938944.....441285917587055633

%e ..877...190899322....470798015742024.6105599904286957450405

%e .4140.10480142147.442649055938121520

%e Some solutions for n=4 and k=3:

%e ..0..1..2....0..0..0....0..0..0....0..0..0....0..0..0....0..0..0....0..0..0

%e ..0..0..1....0..1..0....0..1..0....0..1..0....0..1..2....0..1..0....0..1..0

%e ..0..0..0....0..0..2....2..0..0....2..0..1....1..1..0....0..2..0....0..0..0

%e ..0..1..0....0..0..0....0..2..0....0..0..0....0..0..0....0..0..0....0..2..1

%Y Column 1 and row 1 are A000110.

%Y Column 2 is A020557.

%K nonn,tabl

%O 1,2

%A _R. H. Hardin_, Feb 27 2012