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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.

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`%I #8 Oct 14 2024 15:37:28
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`%S 1,2,2,5,15,5,15,198,203,15,52,4041,20746,4140,52,203,113458,4132120,
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`%T 4150760,115975,203,877,4132120,1358524513,10318694804,1366230232,
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`%U 4213597,877,4140,187612143,671329819215,50996571454200,51074630353994
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`%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.
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`%H R. H. Hardin, <a href="/A208466/b208466.txt">Table of n, a(n) for n = 1..58</a>
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`%e Table starts:
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`%e ....1...........2..................5.....................15
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`%e ....2..........15................198...................4041
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`%e ....5.........203..............20746................4132120
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`%e ...15........4140............4150760............10318694804
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`%e ...52......115975.........1366230232.........51074630353994
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`%e ..203.....4213597.......675203938944.....441285917587055633
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`%e ..877...190899322....470798015742024.6105599904286957450405
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`%e .4140.10480142147.442649055938121520
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`%e Some solutions for n=4 and k=3:
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`%e ..0..1..2....0..0..0....0..0..0....0..0..0....0..0..0....0..0..0....0..0..0
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`%e ..0..0..1....0..1..0....0..1..0....0..1..0....0..1..2....0..1..0....0..1..0
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`%e ..0..0..0....0..0..2....2..0..0....2..0..1....1..1..0....0..2..0....0..0..0
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`%e ..0..1..0....0..0..0....0..2..0....0..0..0....0..0..0....0..0..0....0..2..1
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`%Y Column 1 and row 1 are A000110.
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`%Y Column 2 is A020557.
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`%K nonn,tabl
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`%O 1,2
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`%A _R. H. Hardin_, Feb 27 2012
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