|
%I #11 Nov 25 2015 04:35:11
%S 0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,5,19,33,33,19,5,15,120,413,615,413,120,
%T 15,35,483,2859,6997,6997,2859,483,35,70,1500,13976,53950,84910,53950,
%U 13976,1500,70,126,3923,54199,315198,762227,762227,315198,54199,3923,126,210
%N T(n,k) = Number of n X k nonnegative integer arrays with upper left 0 and lower right n+k-6 and value increasing by 0 or 1 with every step right or down.
%C Table starts
%C ...0....0......0........0.........1...........5...........15............35
%C ...0....0......0........1........19.........120..........483..........1500
%C ...0....0......1.......33.......413........2859........13976.........54199
%C ...0....1.....33......615......6997.......53950.......315198.......1499394
%C ...1...19....413.....6997.....84910......762227......5385305......31454256
%C ...5..120...2859....53950....762227.....8241540.....71297441.....512868867
%C ..15..483..13976...315198...5385305....71297441....759337545....6725497344
%C ..35.1500..54199..1499394..31454256...512868867...6725497344...73117894428
%C ..70.3923.177848..6083808.157376166..3160111147..50869309436..675539536773
%C .126.9069.513905.21733215.692347393.17063990547.335549742230.5411459549576
%H R. H. Hardin, <a href="/A252930/b252930.txt">Table of n, a(n) for n = 1..449</a>
%H R. J. Mathar, <a href="http://vixra.org/abs/1511.0225">Counting 2-way monotonic terrace forms over rectangular landscapes</a>, vixra 1511.0225 (2015)
%F Empirical for column k:
%F k=1: a(n) = (1/24)*n^4 - (5/12)*n^3 + (35/24)*n^2 - (25/12)*n + 1.
%F k=2: [polynomial of degree 8]
%F k=3: [polynomial of degree 12]
%F k=4: [polynomial of degree 16]
%F k=5: [polynomial of degree 20]
%F k=6: [polynomial of degree 24]
%F k=7: [polynomial of degree 28]
%F Empirical: with "n+k-3" instead of "n+k-6" T(n,k) = binomial(n+k,k) - 2.
%e Some solutions for n=4, k=4:
%e ..0..1..1..1....0..0..0..0....0..0..0..0....0..1..1..2....0..0..0..1
%e ..1..1..1..1....1..1..1..1....0..0..0..1....1..1..2..2....0..1..1..2
%e ..1..1..2..2....1..1..1..1....1..1..1..1....1..1..2..2....0..1..1..2
%e ..1..2..2..2....1..1..1..2....1..2..2..2....1..2..2..2....1..2..2..2
%Y Column 1 is A000332(n-1), other columns are A252924 - A252929. Cf. A252923 (diagonal); A252876 (lower right n+k-4), A252976 (lower right n+k-5).
%K nonn,tabl
%O 1,16
%A _R. H. Hardin_, Dec 24 2014
| 