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%I #4 Nov 27 2014 14:00:25
%S 9,18,18,35,34,36,68,62,66,72,133,114,114,130,144,262,214,196,216,258,
%T 288,519,410,344,350,418,514,576,1032,798,622,572,648,820,1026,1152,
%U 2057,1570,1158,962,996,1234,1622,2050,2304,4106,3110,2208,1680,1558,1812
%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 absolute value of x(i,j)-x(i-1,j) in the j direction
%C Table starts
%C ....9...18....35....68...133...262...519..1032..2057..4106...8203..16396..32781
%C ...18...34....62...114...214...410...798..1570..3110..6186..12334..24626..49206
%C ...36...66...114...196...344...622..1158..2208..4284..8410..16634..33052..65856
%C ...72..130...216...350...572...962..1680..3046..5700.10922..21272..41870..82956
%C ..144..258...418...648...996..1558..2526..4284..7600.14010..26586..51472.100956
%C ..288..514...820..1234..1812..2666..4020..6322.10468.18250..33252..62642.120756
%C ..576.1026..1622..2396..3412..4798..6810..9960.15272.24794..42622..76948.144156
%C .1152.2050..3224..4710..6580..8978.12192.16798.23948.35946..57400..97526.174756
%C .2304.4098..6426..9328.12884.17254.22758.30036.40368.56314..82994.130648.219756
%C .4608.8194.12828.18554.25460.33722.43692.56074.72276.95114.130220.188858.293556
%H R. H. Hardin, <a href="/A250769/b250769.txt">Table of n, a(n) for n = 1..611</a>
%F Empirical for column k: (k+2)^2*2^(n-1) plus a linear polynomial in n
%F k=1: a(n) = 2*a(n-1); a(n) = 9*2^(n-1)
%F k=2: a(n) = 3*a(n-1) -2*a(n-2); a(n) = 16*2^(n-1) + 2
%F k=3: a(n) = 4*a(n-1) -5*a(n-2) +2*a(n-3); a(n) = 25*2^(n-1) + 2*n + 8
%F k=4: a(n) = 4*a(n-1) -5*a(n-2) +2*a(n-3); a(n) = 36*2^(n-1) + 10*n + 22
%F k=5: a(n) = 4*a(n-1) -5*a(n-2) +2*a(n-3); a(n) = 49*2^(n-1) + 32*n + 52
%F k=6: a(n) = 4*a(n-1) -5*a(n-2) +2*a(n-3); a(n) = 64*2^(n-1) + 84*n + 114
%F k=7: a(n) = 4*a(n-1) -5*a(n-2) +2*a(n-3); a(n) = 81*2^(n-1) + 198*n + 240
%F k=8: a(n) = 4*a(n-1) -5*a(n-2) +2*a(n-3); a(n) = 100*2^(n-1) + 438*n + 494
%F k=9: a(n) = 4*a(n-1) -5*a(n-2) +2*a(n-3); a(n) = 121*2^(n-1) + 932*n + 1004
%F Empirical for row n: (4*n+4)*2^(k-1) plus a quadratic polynomial in k
%F n=1: a(n) = 4*a(n-1) -5*a(n-2) +2*a(n-3); a(n) = 8*2^(n-1) + n
%F n=2: a(n) = 4*a(n-1) -5*a(n-2) +2*a(n-3); a(n) = 12*2^(n-1) + 4*n + 2
%F n=3: a(n) = 5*a(n-1) -9*a(n-2) +7*a(n-3) -2*a(n-4); a(n) = 16*2^(n-1) + n^2 + 11*n + 8
%F n=4: a(n) = 5*a(n-1) -9*a(n-2) +7*a(n-3) -2*a(n-4); a(n) = 20*2^(n-1) + 4*n^2 + 26*n + 22
%F n=5: a(n) = 5*a(n-1) -9*a(n-2) +7*a(n-3) -2*a(n-4); a(n) = 24*2^(n-1) + 11*n^2 + 57*n + 52
%F n=6: a(n) = 5*a(n-1) -9*a(n-2) +7*a(n-3) -2*a(n-4); a(n) = 28*2^(n-1) + 26*n^2 + 120*n + 114
%F n=7: a(n) = 5*a(n-1) -9*a(n-2) +7*a(n-3) -2*a(n-4); a(n) = 32*2^(n-1) + 57*n^2 + 247*n + 240
%F n=8: a(n) = 5*a(n-1) -9*a(n-2) +7*a(n-3) -2*a(n-4); a(n) = 36*2^(n-1) + 120*n^2 + 502*n + 494
%F n=9: a(n) = 5*a(n-1) -9*a(n-2) +7*a(n-3) -2*a(n-4); a(n) = 40*2^(n-1) + 247*n^2 + 1013*n + 1004
%e Some solutions for n=4 k=4
%e ..1..1..1..1..0....1..0..0..0..0....1..0..1..1..0....1..1..0..1..1
%e ..1..1..1..1..0....1..1..1..1..1....1..0..1..1..0....1..1..0..1..1
%e ..1..1..1..1..0....0..0..0..0..0....1..0..1..1..1....1..1..0..1..1
%e ..1..1..1..1..0....0..0..0..0..1....1..0..1..1..1....1..1..0..1..1
%e ..0..0..0..0..1....0..0..0..0..1....1..0..1..1..1....1..1..0..1..1
%Y Column 1 is A005010(n-1)
%Y Column 2 is A052548(n+3)
%Y Row 1 is A083706(n+1)
%K nonn,tabl
%O 1,1
%A _R. H. Hardin_, Nov 27 2014
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