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%I #4 Nov 28 2014 11:46:10
%S 37,127,125,403,431,413,1229,1325,1450,1333,3673,3867,4388,4750,4213,
%T 10875,11029,12387,14196,15111,13111,32095,31215,33821,39003,44525,
%U 47061,40357,94729,88421,91030,102789,119359,136509,144442,123271,280069
%N T(n,k)=Number of (n+1)X(k+1) 0..2 arrays with nondecreasing x(i,j)-x(i,j-1) in the i direction and nondecreasing absolute value of x(i,j)-x(i-1,j) in the j direction
%C Table starts
%C ......37.....127......403.....1229.....3673.....10875.....32095.....94729
%C .....125.....431.....1325.....3867....11029.....31215.....88421....251819
%C .....413....1450.....4388....12387....33821.....91030....244200....657443
%C ....1333....4750....14196....39003...102789....265546....680504...1742387
%C ....4213...15111....44525...119359...305429....764315...1893433...4673299
%C ...13111...47061...136509...357335...890199...2168405...5232461..12579759
%C ...40357..144442...412006..1052787..2553537...6060594..14283986..33622407
%C ..123271..439056..1231112..3069907..7243359..16732140..38498188..88810083
%C ..374509.1326285..3657629..8903447.20421217..45832201.102731213.231909787
%C .1133503.3990883.10837431.25780959.57483879.125162059.272597111.600517559
%H R. H. Hardin, <a href="/A250898/b250898.txt">Table of n, a(n) for n = 1..180</a>
%F Empirical for column k (k=2 recurrence works also for k=1):
%F k=1: a(n) = 8*a(n-1) -23*a(n-2) +28*a(n-3) -12*a(n-4) for n>6
%F k=2-7: a(n) = 12*a(n-1) -60*a(n-2) +162*a(n-3) -255*a(n-4) +234*a(n-5) -116*a(n-6) +24*a(n-7) for n>9
%F Empirical for row n (n=3 recurrence works also for n=1,2):
%F n=1: a(n) = 9*a(n-1) -31*a(n-2) +51*a(n-3) -40*a(n-4) +12*a(n-5)
%F n=2: a(n) = 10*a(n-1) -40*a(n-2) +82*a(n-3) -91*a(n-4) +52*a(n-5) -12*a(n-6)
%F n=3-7: a(n) = 14*a(n-1) -85*a(n-2) +294*a(n-3) -639*a(n-4) +906*a(n-5) -839*a(n-6) +490*a(n-7) -164*a(n-8) +24*a(n-9)
%e Some solutions for n=4 k=4
%e ..0..0..1..0..1....1..1..0..0..0....1..1..1..1..0....2..2..1..0..0
%e ..0..0..1..0..1....1..1..1..1..1....2..2..2..2..2....1..1..2..1..1
%e ..1..1..2..1..2....1..1..1..1..1....0..0..0..0..0....1..1..2..1..1
%e ..1..1..2..1..2....2..2..2..2..2....0..1..1..1..1....0..0..1..0..0
%e ..1..1..2..1..2....0..0..0..0..0....0..1..1..2..2....0..1..2..2..2
%K nonn,tabl
%O 1,1
%A _R. H. Hardin_, Nov 28 2014
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