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%I #4 Nov 28 2014 10:47:21
%S 90,288,440,678,1456,2002,1328,3442,6812,8736,2306,6728,16262,30360,
%T 37130,3680,11644,31928,73122,131068,155080,5518,18520,55386,144248,
%U 317878,553736,640002,7888,27686,88212,250964,629528,1350002,2304492
%N T(n,k)=Number of (n+1)X(k+1) 0..3 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 .......90.......288.......678......1328.......2306.......3680.......5518
%C ......440......1456......3442......6728......11644......18520......27686
%C .....2002......6812.....16262.....31928......55386......88212.....131982
%C .....8736.....30360.....73122....144248.....250964.....400496.....600070
%C ....37130....131068....317878....629528....1097986....1755220....2633198
%C ...155080....553736...1350002...2681528....4685964....7500960...11264166
%C ...640002...2304492...5640102..11227928...19649066...31484612...47315662
%C ..2619056...9488920..23289922..46440248...81358084..130461616..196169030
%C .10653370..38773148..95366678.190392728..333810066..535577460..805653678
%C .43144920.157554216.388124562.775558328.1360557884.2183825600.3286063846
%H R. H. Hardin, <a href="/A250877/b250877.txt">Table of n, a(n) for n = 1..160</a>
%F Empirical: T(n,k) = (((62/3)*k^3+52*k^2+(130/3)*k+12)*4^n -((45/2)*k^3+(81/2)*k^2+18*k)*3^n +(9*k^3-9*k)*2^n +((5/6)*k^3-(5/2)*k^2+(8/3)*k))/3
%F Empirical for column k:
%F k=1: a(n) = 8*a(n-1) -19*a(n-2) +12*a(n-3); a(n) = (128*4^n-81*3^n+1)/3
%F k=2: a(n) = 10*a(n-1) -35*a(n-2) +50*a(n-3) -24*a(n-4); a(n) = (472*4^n-378*3^n+54*2^n+2)/3
%F k=3: a(n) = 10*a(n-1) -35*a(n-2) +50*a(n-3) -24*a(n-4); a(n) = (1168*4^n-1026*3^n+216*2^n+8)/3
%F k=4: a(n) = 10*a(n-1) -35*a(n-2) +50*a(n-3) -24*a(n-4); a(n) = (2340*4^n-2160*3^n+540*2^n+24)/3
%F k=5: a(n) = 10*a(n-1) -35*a(n-2) +50*a(n-3) -24*a(n-4); a(n) = (4112*4^n-3915*3^n+1080*2^n+55)/3
%F k=6: a(n) = 10*a(n-1) -35*a(n-2) +50*a(n-3) -24*a(n-4); a(n) = (6608*4^n-6426*3^n+1890*2^n+106)/3
%F k=7: a(n) = 10*a(n-1) -35*a(n-2) +50*a(n-3) -24*a(n-4); a(n) = (9952*4^n-9828*3^n+3024*2^n+182)/3
%F Empirical for row n:
%F n=1: a(n) = (34/3)*n^3 + 28*n^2 + (104/3)*n + 16
%F n=2: a(n) = 55*n^3 + 155*n^2 + 166*n + 64
%F n=3: a(n) = (788/3)*n^3 + 744*n^2 + (2218/3)*n + 256
%F n=4: a(n) = (3613/3)*n^3 + 3343*n^2 + (9494/3)*n + 1024
%F n=5: a(n) = 5328*n^3 + 14468*n^2 + 13238*n + 4096
%F n=6: a(n) = (68825/3)*n^3 + 61155*n^2 + (163798/3)*n + 16384
%F n=7: a(n) = (290548/3)*n^3 + 254464*n^2 + (669458/3)*n + 65536
%e Some solutions for n=4 k=4
%e ..0..0..0..0..1....0..0..0..1..1....0..0..0..0..0....1..1..1..1..1
%e ..2..2..2..2..3....2..2..2..3..3....2..2..3..3..3....0..0..0..0..0
%e ..2..2..2..2..3....1..1..2..3..3....1..1..2..3..3....2..2..2..2..2
%e ..0..1..1..1..2....0..0..1..2..2....1..1..2..3..3....0..2..2..2..2
%e ..0..1..1..1..3....0..0..2..3..3....0..0..1..3..3....0..2..2..3..3
%K nonn,tabl
%O 1,1
%A _R. H. Hardin_, Nov 28 2014
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