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%I #4 Nov 26 2014 09:30:19
%S 11,26,26,57,64,57,120,140,140,120,247,290,297,290,247,502,586,592,
%T 592,586,502,1013,1172,1153,1126,1153,1172,1013,2036,2336,2236,2092,
%U 2092,2236,2336,2036,4083,4654,4353,3890,3691,3890,4353,4654,4083,8178,9278
%N T(n,k)=Number of (n+1)X(k+1) 0..1 arrays with nondecreasing x(i,j)-x(i,j-1) in the i direction and nondecreasing x(i,j)-x(i-1,j) in the j direction
%C Table starts
%C ...11....26....57...120...247....502...1013...2036...4083...8178..16369...32752
%C ...26....64...140...290...586...1172...2336...4654...9278..18512..36964...73850
%C ...57...140...297...592..1153...2236...4353...8528..16809..33292..66169..131824
%C ..120...290...592..1126..2092...3890...7320..13982..27076..53002.104560..207350
%C ..247...586..1153..2092..3691...6526..11749..21664..40879..78610.153289..301780
%C ..502..1172..2236..3890..6526..10928..18664..32870..59818.112052.214660..417818
%C .1013..2336..4353..7320.11749..18664..30113..50192..87093.157200.293281..560872
%C .2036..4654..8528.13982.21664..32870..50192..78814.129104.221798.398368..741758
%C .4083..9278.16809.27076.40879..59818..87093.129104.198651.321334.548353..982108
%C .8178.18512.33292.53002.78610.112052.157200.221798.321334.486784.780100.1325186
%H R. H. Hardin, <a href="/A250611/b250611.txt">Table of n, a(n) for n = 1..881</a>
%F Empirical: T(n,k) = 2^(n-1)*((k+1)*2)^2 + a quadratic polynomial in n
%F Empirical for column k (k=2 recurrence also works for k=1):
%F k=1: a(n) = 4*a(n-1) -5*a(n-2) +2*a(n-3); a(n)=16*2^(n-1) -n-4
%F k=2: a(n) = 5*a(n-1) -9*a(n-2) +7*a(n-3) -2*a(n-4); a(n)=36*2^(n-1) +n^2-n-10
%F k=3: a(n) = 5*a(n-1) -9*a(n-2) +7*a(n-3) -2*a(n-4); a(n)=64*2^(n-1) +5*n^2+4*n-16
%F k=4: a(n) = 5*a(n-1) -9*a(n-2) +7*a(n-3) -2*a(n-4); a(n)=100*2^(n-1) +16*n^2+22*n-18
%F k=5: a(n) = 5*a(n-1) -9*a(n-2) +7*a(n-3) -2*a(n-4); a(n)=144*2^(n-1) +42*n^2+69*n-8
%F k=6: a(n) = 5*a(n-1) -9*a(n-2) +7*a(n-3) -2*a(n-4); a(n)=196*2^(n-1) +99*n^2+177*n+30
%F k=7: a(n) = 5*a(n-1) -9*a(n-2) +7*a(n-3) -2*a(n-4); a(n)=256*2^(n-1) +219*n^2+410*n+128
%F Empirical for diagonal: a(n) = 11*a(n-1) -52*a(n-2) +138*a(n-3) -225*a(n-4) +231*a(n-5) -146*a(n-6) +52*a(n-7) -8*a(n-8)
%e Some solutions for n=6 k=4
%e ..1..1..0..0..0....1..1..1..0..0....1..1..1..1..0....0..0..0..1..0
%e ..1..1..0..0..0....1..1..1..1..1....1..1..1..1..0....0..0..0..1..0
%e ..1..1..0..0..0....1..1..1..1..1....1..1..1..1..0....0..0..0..1..0
%e ..0..0..0..0..0....1..1..1..1..1....1..1..1..1..0....0..0..0..1..0
%e ..0..1..1..1..1....1..1..1..1..1....1..1..1..1..0....0..0..0..1..0
%e ..0..1..1..1..1....0..0..0..1..1....1..1..1..1..0....0..0..0..1..0
%e ..0..1..1..1..1....0..0..0..1..1....0..1..1..1..1....0..0..0..1..0
%Y Column 1 is A000295(n+3)
%K nonn,tabl
%O 1,1
%A _R. H. Hardin_, Nov 26 2014
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