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T(n,k)=Number of nXk 0..1 arrays with rows and columns in lexicographic nondecreasing order but with exactly one mistake.
8

%I #4 Nov 19 2016 12:02:49

%S 0,1,1,4,6,4,10,27,27,10,20,96,142,96,20,35,281,701,701,281,35,56,708,

%T 3183,4872,3183,708,56,84,1590,12875,34038,34038,12875,1590,84,120,

%U 3264,46236,229887,366982,229887,46236,3264,120,165,6237,149099,1429751

%N T(n,k)=Number of nXk 0..1 arrays with rows and columns in lexicographic nondecreasing order but with exactly one mistake.

%C Table starts

%C ...0.....1.......4........10..........20............35...............56

%C ...1.....6......27........96.........281...........708.............1590

%C ...4....27.....142.......701........3183.........12875............46236

%C ..10....96.....701......4872.......34038........229887..........1429751

%C ..20...281....3183.....34038......366982.......4058169.........44208465

%C ..35...708...12875....229887.....4058169......72941420.......1349344763

%C ..56..1590...46236...1429751....44208465....1349344763......41820621848

%C ..84..3264..149099...8023290...451571539...24833319003....1335072188627

%C .120..6237..438091..40628537..4213513472..434369879650...42737962734110

%C .165.11242.1188762.187182375.35702217342.7020524920612.1315545171895930

%H R. H. Hardin, <a href="/A278363/b278363.txt">Table of n, a(n) for n = 1..199</a>

%F Empirical for column k:

%F k=1: a(n) = (1/6)*n^3 - (1/6)*n

%F k=2: [polynomial of degree 7]

%F k=3: [polynomial of degree 15]

%F k=4: [polynomial of degree 31]

%F k=5: [polynomial of degree 63]

%F k=6: [polynomial of degree 127]

%e Some solutions for n=4 k=4

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

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

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

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

%Y Column 1 is A000292(n-1).

%K nonn,tabl

%O 1,4

%A _R. H. Hardin_, Nov 19 2016