%I #4 Jun 03 2015 09:14:39
%S 16,44,44,104,112,104,228,210,296,228,480,344,598,652,480,988,506,985,
%T 1244,1329,988,2008,683,1403,1891,2384,2530,2008,4052,894,1917,2509,
%U 3754,3980,4667,4052,8144,1140,2542,3314,5070,6196,6348,8419,8144,16332,1421
%N T(n,k)=Number of (n+1)X(k+1) 0..1 arrays with every 2X2 subblock ne-sw antidiagonal difference nondecreasing horizontally and nw+se diagonal sum nondecreasing vertically
%C Table starts
%C ....16....44...104...228...480...988..2008..4052..8144..16332..32712..65476
%C ....44...112...210...344...506...683...894..1140..1421...1725...2071...2460
%C ...104...296...598...985..1403..1917..2542..3222..3940...4767...5718...6737
%C ...228...652..1244..1891..2509..3314..4313..5356..6331...7470...8820..10231
%C ...480..1329..2384..3754..5070..6662..8612.10720.12831..15045..17532..20200
%C ...988..2530..3980..6196..8339.10854.13833.17058.20414..23858..27446..31237
%C ..2008..4667..6348..9688.12940.16942.21653.26628.31806..37351..43158..49004
%C ..4052..8419..9567.14216.18630.24300.31254.38451.45713..53599..62205..70891
%C ..8144.14932.13847.20285.26240.33844.43655.54137.64507..75625..87878.100660
%C .16332.26184.19227.27687.35386.45166.57928.71823.85853.100830.116939.133789
%H R. H. Hardin, <a href="/A258554/b258554.txt">Table of n, a(n) for n = 1..8053</a>
%F Empirical for column k:
%F k=1: a(n) = 4*a(n-1) -5*a(n-2) +2*a(n-3)
%F k=2: a(n) = 4*a(n-1) -5*a(n-2) +a(n-3) +3*a(n-4) -4*a(n-5) +2*a(n-6) +2*a(n-7) -3*a(n-8) +a(n-9) for n>11
%F k=3: a(n) = 4*a(n-1) -6*a(n-2) +4*a(n-3) -4*a(n-5) +6*a(n-6) -4*a(n-7) +a(n-8) for n>11
%F k=4: a(n) = 4*a(n-1) -6*a(n-2) +4*a(n-3) -4*a(n-5) +6*a(n-6) -4*a(n-7) +a(n-8) for n>12
%F k=5: a(n) = 4*a(n-1) -6*a(n-2) +4*a(n-3) -4*a(n-5) +6*a(n-6) -4*a(n-7) +a(n-8) for n>13
%F k=6: a(n) = 4*a(n-1) -6*a(n-2) +4*a(n-3) -4*a(n-5) +6*a(n-6) -4*a(n-7) +a(n-8) for n>14
%F k=7: a(n) = 4*a(n-1) -6*a(n-2) +4*a(n-3) -4*a(n-5) +6*a(n-6) -4*a(n-7) +a(n-8) for n>15
%F Empirical for row n:
%F n=1: a(n) = 4*a(n-1) -5*a(n-2) +2*a(n-3)
%F n=2: a(n) = 3*a(n-1) -3*a(n-2) +a(n-3) +a(n-4) -3*a(n-5) +3*a(n-6) -a(n-7) for n>10
%F n=3: a(n) = 3*a(n-1) -3*a(n-2) +a(n-3) +a(n-4) -3*a(n-5) +3*a(n-6) -a(n-7) for n>10
%F n=4: a(n) = 3*a(n-1) -3*a(n-2) +a(n-3) +a(n-4) -3*a(n-5) +3*a(n-6) -a(n-7) for n>11
%F n=5: a(n) = 3*a(n-1) -3*a(n-2) +a(n-3) +a(n-4) -3*a(n-5) +3*a(n-6) -a(n-7) for n>12
%F n=6: a(n) = 3*a(n-1) -3*a(n-2) +a(n-3) +a(n-4) -3*a(n-5) +3*a(n-6) -a(n-7) for n>13
%F n=7: a(n) = 3*a(n-1) -3*a(n-2) +a(n-3) +a(n-4) -3*a(n-5) +3*a(n-6) -a(n-7) for n>14
%e Some solutions for n=4 k=4
%e ..1..0..0..0..1....0..0..0..0..1....1..0..0..0..0....0..0..0..0..0
%e ..1..0..0..0..1....0..0..0..0..0....1..0..0..0..0....1..0..0..0..0
%e ..1..0..0..0..1....1..0..0..0..0....0..0..0..0..0....0..0..0..0..0
%e ..1..0..0..1..1....1..1..1..1..1....1..1..0..0..0....1..1..1..1..1
%e ..1..1..1..1..1....1..1..1..1..0....1..0..0..0..0....0..0..0..0..0
%K nonn,tabl
%O 1,1
%A _R. H. Hardin_, Jun 03 2015