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T(n,k)=Number of nXk 0..4 arrays of sums of 2X2 subblocks of some (n+1)X(k+1) binary array with rows and columns of the latter in lexicographically nondecreasing order
6

%I #5 Jun 25 2013 21:04:29

%S 5,10,10,15,42,15,21,120,120,21,28,313,608,313,28,36,729,2820,2820,

%T 729,36,45,1556,11325,24158,11325,1556,45,55,3099,40431,180712,180712,

%U 40431,3099,55,66,5818,130479,1187869,2601925,1187869,130479,5818,66,78,10384

%N T(n,k)=Number of nXk 0..4 arrays of sums of 2X2 subblocks of some (n+1)X(k+1) binary array with rows and columns of the latter in lexicographically nondecreasing order

%C Table starts

%C ..5....10......15........21..........28............36...............45

%C .10....42.....120.......313.........729..........1556.............3099

%C .15...120.....608......2820.......11325.........40431...........130479

%C .21...313....2820.....24158......180712.......1187869..........6897735

%C .28...729...11325....180712.....2601925......33118416........369163839

%C .36..1556...40431...1187869....33118416.....833294380......18415822936

%C .45..3099..130479...6897735...369163839...18415822936.....816606210291

%C .55..5818..385529..35818171..3630637294..357313893357...31841532264308

%C .66.10384.1054857.168412446.31856129416.6135150314839.1095361466285336

%H R. H. Hardin, <a href="/A226992/b226992.txt">Table of n, a(n) for n = 1..143</a>

%F Empirical for column k:

%F k=1: a(n) = (1/2)*n^2 + (5/2)*n + 3 for n>1

%F k=2: [polynomial of degree 7] for n>4

%F k=3: [polynomial of degree 15] for n>7

%F k=4: [polynomial of degree 31] for n>15

%e Some solutions for n=4 k=4

%e ..0..1..2..3....0..0..1..3....0..1..3..4....0..1..2..3....0..1..2..3

%e ..1..3..4..4....0..1..3..3....1..2..2..3....2..2..3..4....1..3..4..3

%e ..2..4..4..4....1..2..3..2....2..1..0..1....4..3..2..2....3..3..2..2

%e ..2..2..3..4....2..1..1..2....2..0..0..0....4..4..3..2....4..3..2..3

%K nonn,tabl

%O 1,1

%A _R. H. Hardin_ Jun 25 2013