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A186180 T(n,k)=Number of (n+2)X(k+2) 0..5 arrays with each 3X3 subblock having rows and columns in lexicographically nondecreasing order.

%I A186180 #10 Jul 22 2025 10:18:46
%S A186180 520017,10084236,10084236,143369699,311128593,143369699,1662436696,
%T A186180 6520730198,6520730198,1662436696,16382439469,105970767207,
%U A186180 188034884094,105970767207,16382439469,140871930232,1414199542732,4041778238254
%N A186180 T(n,k)=Number of (n+2)X(k+2) 0..5 arrays with each 3X3 subblock having rows and columns in lexicographically nondecreasing order.
%C A186180 Table starts
%C A186180 ..........520017..........10084236............143369699............1662436696
%C A186180 ........10084236.........311128593...........6520730198..........105970767207
%C A186180 .......143369699........6520730198.........188034884094.........4041778238254
%C A186180 ......1662436696......105970767207........4041778238254.......111203560772547
%C A186180 .....16382439469.....1414199542732.......69471558136868......2391923493659465
%C A186180 ....140871930232....16059530994398......995828085723859.....42174821764604242
%C A186180 ...1078197169699...159099595031390....12251749347425002....629512200937395977
%C A186180 ...7459396065112..1400823449171621...132151619698400257...8143852416376007571
%C A186180 ..47221234070168.11121210203531892..1270399513311212137..92981285763140685886
%C A186180 .276218909139304.80539662788823416.11027904404610778911.950506396177707075676
%H A186180 R. H. Hardin, <a href="/A186180/b186180.txt">Table of n, a(n) for n = 1..178</a>
%H A186180 R. H. Hardin, <a href="/A186180/a186180.txt">Polynomials for columns 1-5</a>
%F A186180 Empirical: T(n,k) is a polynomial of degree 5k+50 in n, for fixed k.
%F A186180 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 A186180 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 A186180 Some solutions for 5X4
%e A186180 ..0..0..0..0....0..0..0..0....0..0..0..0....0..0..0..0....0..0..0..0
%e A186180 ..0..0..0..0....0..0..0..0....0..0..0..0....0..0..0..0....0..0..0..0
%e A186180 ..0..0..0..3....0..0..0..0....0..0..0..0....0..0..0..3....0..0..0..0
%e A186180 ..0..0..0..5....0..0..1..2....0..1..1..4....0..1..5..1....0..0..2..3
%e A186180 ..0..1..1..0....1..2..0..2....3..1..4..1....5..4..4..5....0..2..5..1
%K A186180 nonn,tabl
%O A186180 1,1
%A A186180 _R. H. Hardin_, General degree formula intuited by _D. S. McNeil_ in the Sequence Fans Mailing List, Feb 13 2011