%I #4 Nov 27 2014 11:25:45

%S 32,72,105,129,237,332,203,423,756,1029,294,663,1353,2361,3152,402,

%T 957,2123,4239,7272,9585,527,1305,3066,6663,13089,22197,29012,669,

%U 1707,4182,9633,20603,40023,67356,87549,828,2163,5471,13149,29814,63063,121593

%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 x(i,j)+x(i-1,j) in the j direction

%C Table starts

%C .....32......72.....129.....203.....294......402......527......669......828

%C ....105.....237.....423.....663.....957.....1305.....1707.....2163.....2673

%C ....332.....756....1353....2123....3066.....4182.....5471.....6933.....8568

%C ...1029....2361....4239....6663....9633....13149....17211....21819....26973

%C ...3152....7272...13089...20603...29814....40722....53327....67629....83628

%C ...9585...22197...40023...63063...91317...124785...163467...207363...256473

%C ..29012...67356..121593..191723..277746...379662...497471...631173...780768

%C ..87549..203601..367839..580263..840873..1149669..1506651..1911819..2365173

%C .263672..613872.1109649.1751003.2537934..3470442..4548527..5772189..7141428

%C .793065.1847757.3341223.5273463.7644477.10454265.13702827.17390163.21516273

%H R. H. Hardin, <a href="/A250755/b250755.txt">Table of n, a(n) for n = 1..241</a>

%F Empirical: T(n,k) = (3*(k+1)*(5*k+4)*3^n - (8*k^2+8*k)*2^n + (5*k^2-7*k))/4

%F Empirical for column k:

%F k=1: a(n) = 6*a(n-1) -11*a(n-2) +6*a(n-3); a(n) = (27*3^n-8*2^n-1)/2

%F k=2: a(n) = 6*a(n-1) -11*a(n-2) +6*a(n-3); a(n) = (63*3^n-24*2^n+3)/2

%F k=3: a(n) = 6*a(n-1) -11*a(n-2) +6*a(n-3); a(n) = (114*3^n-48*2^n+12)/2

%F k=4: a(n) = 6*a(n-1) -11*a(n-2) +6*a(n-3); a(n) = (180*3^n-80*2^n+26)/2

%F k=5: a(n) = 6*a(n-1) -11*a(n-2) +6*a(n-3); a(n) = (261*3^n-120*2^n+45)/2

%F k=6: a(n) = 6*a(n-1) -11*a(n-2) +6*a(n-3); a(n) = (357*3^n-168*2^n+69)/2

%F k=7: a(n) = 6*a(n-1) -11*a(n-2) +6*a(n-3); a(n) = (468*3^n-224*2^n+98)/2

%F Empirical for row n:

%F n=1: a(n) = (17/2)*n^2 + (29/2)*n + 9

%F n=2: a(n) = 27*n^2 + 51*n + 27

%F n=3: a(n) = (173/2)*n^2 + (329/2)*n + 81

%F n=4: a(n) = 273*n^2 + 513*n + 243

%F n=5: a(n) = (1697/2)*n^2 + (3149/2)*n + 729

%F n=6: a(n) = 2607*n^2 + 4791*n + 2187

%F n=7: a(n) = (15893/2)*n^2 + (29009/2)*n + 6561

%e Some solutions for n=4 k=4

%e ..1..1..1..1..0....0..0..0..0..0....0..0..0..0..0....0..0..0..0..0

%e ..1..1..1..1..2....2..2..2..2..2....1..1..1..1..1....0..0..0..0..0

%e ..1..1..1..1..2....0..0..0..0..0....0..0..0..0..0....1..1..1..1..1

%e ..0..1..1..1..2....1..1..1..2..2....2..2..2..2..2....0..0..2..2..2

%e ..0..1..1..1..2....0..0..0..1..2....0..1..1..1..2....0..0..2..2..2

%Y Column 1 is A053152(n+3)

%K nonn,tabl

%O 1,1

%A _R. H. Hardin_, Nov 27 2014