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A186096 T(n,k)=Number of (n+2)X(k+2) 0..4 arrays with each 3X3 subblock having rows and columns in lexicographically nondecreasing order.

%I A186096 #14 Jul 22 2025 10:15:02
%S A186096 102251,1252889,1252889,11258613,22559052,11258613,83378583,280102672,
%T A186096 280102672,83378583,531218757,2743553694,4527262140,2743553694,
%U A186096 531218757,2985984444,22408644868,55707179395,55707179395,22408644868,2985984444
%N A186096 T(n,k)=Number of (n+2)X(k+2) 0..4 arrays with each 3X3 subblock having rows and columns in lexicographically nondecreasing order.
%C A186096 Table starts
%C A186096 ........102251.........1252889.........11258613...........83378583
%C A186096 .......1252889........22559052........280102672.........2743553694
%C A186096 ......11258613.......280102672.......4527262140........55707179395
%C A186096 ......83378583......2743553694......55707179395.......837192826927
%C A186096 .....531218757.....22408644868.....558643720724.....10064164793382
%C A186096 ....2985984444....157927508610....4754203179765....101247852066065
%C A186096 ...15084070635....983600385660...35285910378578....878623899164100
%C A186096 ...69482992431...5510351270895..232998389350277...6723402580436327
%C A186096 ..295278398390..28148281162513.1389861134920751..46135247077059665
%C A186096 .1168636004931.132536596243411.7581135805604097.287649593317228144
%H A186096 R. H. Hardin, <a href="/A186096/b186096.txt">Table of n, a(n) for n = 1..219</a>
%H A186096 R. H. Hardin, <a href="/A186096/a186096.txt">Polynomials for columns 1-8</a>
%F A186096 Empirical: T(n,k) is a polynomial of degree 4k+30 in n, for fixed k.
%F A186096 Let T(n,k,z) be the number of (n+2)X(k+2) 0..z arrays with each 3X3 subblock having rows and columns in lexicographically nondecreasing order.
%F A186096 Then empirically T(n,k,z) is a polynomial of degree z*k + z*(z+1)*(z+5)/6 in n, for fixed k.
%e A186096 Some solutions for 5X4
%e A186096 ..0..0..0..0....0..0..0..0....0..0..0..0....0..0..0..0....0..0..0..0
%e A186096 ..0..0..0..0....0..0..0..0....0..0..0..0....0..0..0..0....0..0..0..0
%e A186096 ..0..0..0..2....0..0..0..2....0..0..0..2....0..0..1..2....0..0..1..2
%e A186096 ..0..1..2..1....1..1..2..2....1..1..2..0....1..2..4..4....0..2..1..2
%e A186096 ..2..3..3..4....1..2..0..0....3..4..0..1....1..4..1..3....2..4..3..2
%K A186096 nonn,tabl
%O A186096 1,1
%A A186096 _R. H. Hardin_, General degree formula intuited by _D. S. McNeil_ in the Sequence Fans Mailing List, Feb 12 2011