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A250783 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.

%I A250783 #6 Jul 23 2025 12:45:50
%S A250783 9,21,18,46,46,36,99,106,96,72,209,238,230,196,144,436,518,534,482,
%T A250783 396,288,901,1106,1194,1152,990,796,576,1849,2326,2604,2640,2426,2010,
%U A250783 1596,1152,3774,4838,5568,5882,5688,5028,4054,3196,2304,7671,9978,11732,12796
%N A250783 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 A250783 Table starts
%C A250783 ....9....21....46....99....209....436.....901....1849....3774.....7671....15541
%C A250783 ...18....46...106...238....518...1106....2326....4838....9978....20446....41686
%C A250783 ...36....96...230...534...1194...2604....5568...11732...24442....50482...103566
%C A250783 ...72...196...482..1152...2640...5882...12796...27344...57610...120060...248072
%C A250783 ..144...396...990..2426...5688..12950...28692...62274..132890...279864...583196
%C A250783 ..288...796..2010..5028..12036..27986...63184..139436..301786...643164..1353544
%C A250783 ..576..1596..4054.10306..25126..59590..137082..307762..676266..1460260..3107536
%C A250783 .1152..3196..8146.20960..51904.125334..293588..670608.1496970..3278004..7061504
%C A250783 .2304..6396.16334.42394.106344.260916..621664.1444162.3275574..7278104.15884220
%C A250783 .4608.12796.32714.85420.216500.538538.1303276.3076788.7089558.15987988.35370676
%H A250783 R. H. Hardin, <a href="/A250783/b250783.txt">Table of n, a(n) for n = 1..544</a>
%F A250783 Empirical for column k:
%F A250783 k=1: a(n) = 2*a(n-1); a(n) = 9*2^(n-1)
%F A250783 k=2: a(n) = 3*a(n-1) -2*a(n-2); a(n) = 25*2^(n-1) -4
%F A250783 k=3: a(n) = 4*a(n-1) -5*a(n-2) +2*a(n-3)
%F A250783 k=4: a(n) = 6*a(n-1) -14*a(n-2) +16*a(n-3) -9*a(n-4) +2*a(n-5)
%F A250783 k=5: a(n) = 8*a(n-1) -27*a(n-2) +50*a(n-3) -55*a(n-4) +36*a(n-5) -13*a(n-6) +2*a(n-7)
%F A250783 k=6: [order 9]
%F A250783 k=7: [order 11]
%F A250783 Empirical for row n:
%F A250783 n=1: a(n) = 3*a(n-1) -a(n-2) -2*a(n-3)
%F A250783 n=2: a(n) = 4*a(n-1) -4*a(n-2) -a(n-3) +2*a(n-4)
%F A250783 n=3: a(n) = 5*a(n-1) -8*a(n-2) +3*a(n-3) +3*a(n-4) -2*a(n-5)
%F A250783 n=4: a(n) = 5*a(n-1) -7*a(n-2) -2*a(n-3) +11*a(n-4) -5*a(n-5) -3*a(n-6) +2*a(n-7)
%F A250783 n=5: [order 8]
%F A250783 n=6: [order 9]
%F A250783 n=7: [order 10]
%e A250783 Some solutions for n=4 k=4
%e A250783 ..0..0..1..0..0....0..0..0..1..0....0..0..0..0..0....1..0..1..1..0
%e A250783 ..0..0..1..0..0....0..0..0..1..1....0..0..0..0..0....1..0..1..1..0
%e A250783 ..0..0..1..0..0....0..0..0..1..1....0..0..0..0..1....1..0..1..1..1
%e A250783 ..0..0..1..0..1....0..0..0..1..1....1..1..1..1..0....1..0..1..1..1
%e A250783 ..0..0..1..0..1....0..0..0..1..1....1..1..1..1..0....1..0..1..1..1
%Y A250783 Column 1 is A005010(n-1)
%Y A250783 Row 1 is A027973(n+1)
%K A250783 nonn,tabl
%O A250783 1,1
%A A250783 _R. H. Hardin_, Nov 27 2014