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T(n,k)=Number of nXk nonnegative integer arrays with each row and column increasing from zero by 0, 1, 2 or 3
7

%I #7 Mar 31 2012 12:36:52

%S 1,1,1,1,4,1,1,10,10,1,1,20,90,20,1,1,35,534,534,35,1,1,56,2310,11016,

%T 2310,56,1,1,84,8012,150590,150590,8012,84,1,1,120,23661,1441046,

%U 7083180,1441046,23661,120,1,1,165,61830,10457226,220242352,220242352,10457226

%N T(n,k)=Number of nXk nonnegative integer arrays with each row and column increasing from zero by 0, 1, 2 or 3

%C Table starts

%C .1...1......1..........1.............1.................1.....................1

%C .1...4.....10.........20............35................56....................84

%C .1..10.....90........534..........2310..............8012.................23661

%C .1..20....534......11016........150590...........1441046..............10457226

%C .1..35...2310.....150590.......7083180.........220242352............4694959782

%C .1..56...8012....1441046.....220242352.......23491810780.........1652619440763

%C .1..84..23661...10457226....4694959782.....1652619440763.......398328574277322

%C .1.120..61830...61213311...73210175188....78726503296249.....63219027201690907

%C .1.165.146718..302092215..884548287930..2676714904101546...6740977636445383462

%C .1.220.321970.1297497783.8662839281016.68406213820923245.504159442506496670736

%H R. H. Hardin, <a href="/A202924/b202924.txt">Table of n, a(n) for n = 1..220</a>

%F Empirical: columns of T(n,k) are polynomials in n of degree 3*k*(k-1)/2.

%F For elements increasing by 0..d instead of 0..3, columns are a polynomial of degree d*k*(k-1)/2.

%e Some solutions for n=5 k=3

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

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

%e ..0..3..3....0..0..2....0..0..1....0..3..3....0..0..3....0..2..2....0..1..3

%e ..0..3..3....0..1..3....0..1..2....0..3..3....0..2..3....0..2..5....0..3..3

%e ..0..3..4....0..2..4....0..2..4....0..3..5....0..3..5....0..3..6....0..3..3

%Y Column 2 is A000292

%K nonn,tabl

%O 1,5

%A _R. H. Hardin_ Dec 26 2011