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A184574 T(n,k)=Number of (n+2)X(k+2) 0..3 arrays with each 3X3 subblock having rows and columns in lexicographically nondecreasing order.

%I A184574 #10 Jul 22 2025 09:38:29
%S A184574 14178,102445,102445,545662,993538,545662,2430950,6803631,6803631,
%T A184574 2430950,9496395,37767705,57374460,37767705,9496395,33351260,
%U A184574 179122657,380532059,380532059,179122657,33351260,107058241,748499580,2113138210
%N A184574 T(n,k)=Number of (n+2)X(k+2) 0..3 arrays with each 3X3 subblock having rows and columns in lexicographically nondecreasing order.
%C A184574 Table starts
%C A184574 ......14178......102445........545662........2430950.........9496395
%C A184574 .....102445......993538.......6803631.......37767705.......179122657
%C A184574 .....545662.....6803631......57374460......380532059......2113138210
%C A184574 ....2430950....37767705.....380532059.....2943827443.....18803004899
%C A184574 ....9496395...179122657....2113138210....18803004899....136680720320
%C A184574 ...33351260...748499580...10202200416...103444456133....848542379467
%C A184574 ..107058241..2816118529...43935544294...503785839330...4626643143791
%C A184574 ..318063303..9696377100..171891306894..2213469458762..22587829272879
%C A184574 ..883398416.30941723282..619309263773..8896632071640.100176548344077
%C A184574 .2312834051.92420016377.2076328840978.33064363109286.408222405584237
%H A184574 R. H. Hardin, <a href="/A184574/b184574.txt">Table of n, a(n) for n = 1..480</a>
%H A184574 R. H. Hardin, <a href="/A184574/a184574.txt">Polynomials for columns 1-8</a>
%F A184574 Empirical: T(n,k) is a polynomial of degree 3k+16 in n, for fixed k.
%F A184574 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 A184574 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 A184574 Some solutions for 5X4
%e A184574 ..0..0..0..0....0..0..0..0....0..0..0..0....0..0..0..0....0..0..0..0
%e A184574 ..0..0..1..1....0..0..0..1....0..0..1..1....0..0..0..0....0..0..0..0
%e A184574 ..0..0..1..2....0..0..1..1....0..0..1..1....0..0..0..2....0..0..1..2
%e A184574 ..2..3..0..0....0..0..3..3....0..3..1..3....1..2..2..2....0..2..3..2
%e A184574 ..2..3..0..1....0..1..2..2....1..3..3..3....3..2..3..0....0..3..3..1
%K A184574 nonn,tabl
%O A184574 1,1
%A A184574 _R. H. Hardin_, general degree formula intuited by _D. S. McNeil_ in the Sequence Fans Mailing List, Jan 17 2011