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T(n,k) = Number of n X k nonnegative integer arrays with each row and column an ascent sequence (interior element no greater than one plus up-steps preceding it).
6

%I #7 May 09 2022 00:41:22

%S 1,1,1,1,2,1,1,4,4,1,1,8,18,8,1,1,16,86,86,16,1,1,32,422,1094,422,32,

%T 1,1,64,2094,15184,15184,2094,64,1,1,128,10438,219934,658492,219934,

%U 10438,128,1,1,256,52126,3249298,31670778,31670778,3249298,52126,256,1,1,512

%N T(n,k) = Number of n X k nonnegative integer arrays with each row and column an ascent sequence (interior element no greater than one plus up-steps preceding it).

%C Table starts

%C .1...1.....1........1...........1...............1..................1

%C .1...2.....4........8..........16..............32.................64

%C .1...4....18.......86.........422............2094..............10438

%C .1...8....86.....1094.......15184..........219934............3249298

%C .1..16...422....15184......658492........31670778.........1605067272

%C .1..32..2094...219934....31670778......5534917394......1078490778110

%C .1..64.10438..3249298..1605067272...1078490778110....876480920594230

%C .1.128.52126.48427802.83391232150.222425177005132.794166642153146254

%H R. H. Hardin, <a href="/A202549/b202549.txt">Table of n, a(n) for n = 1..144</a>

%e Some solutions for n=4, k=4

%e ..0..0..0..0....0..0..0..0....0..0..0..0....0..0..0..0....0..0..0..0

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

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

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

%Y Column 2 is A000079(n-1).

%Y Column 3 is A082685(n-1).

%K nonn,tabl

%O 1,5

%A _R. H. Hardin_, Dec 20 2011