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%I #8 Jun 08 2018 13:51:18
%S 80,137,284,571,1076,1918,3261,5329,8408,12867,19162,27859,39640,
%T 55328,75895,102489,136444,179309,232860,299131,380428,479362,598865,
%U 742225,913104,1115575,1354142,1633779,1959952,2338660,2776459,3280505,3858580
%N Number of (n+1) X 6 0..1 arrays with column and row pair sums b(i,j)=a(i,j)+a(i,j-1) and c(i,j)=a(i,j)+a(i-1,j) nondecreasing in column and row directions, respectively.
%C Column 5 of A204651.
%H R. H. Hardin, <a href="/A204648/b204648.txt">Table of n, a(n) for n = 1..210</a>
%F Empirical: a(n) = 6*a(n-1) -14*a(n-2) +14*a(n-3) -14*a(n-5) +14*a(n-6) -6*a(n-7) +a(n-8) for n>11.
%F Conjectures from _Colin Barker_, Jun 08 2018: (Start)
%F G.f.: x*(80 - 343*x + 582*x^2 - 335*x^3 - 292*x^4 + 600*x^5 - 379*x^6 + 89*x^7 - 4*x^8 + 8*x^9 - 4*x^10) / ((1 - x)^7*(1 + x)).
%F a(n) = (-10080 + 43776*n + 15308*n^2 + 2970*n^3 + 620*n^4 + 54*n^5 + 2*n^6)/1440 for n>3 and even.
%F a(n) = (-10890 + 43776*n + 15308*n^2 + 2970*n^3 + 620*n^4 + 54*n^5 + 2*n^6)/1440 for n>3 and odd.
%F (End)
%e Some solutions for n=5:
%e ..1..0..1..0..1..0....0..0..0..0..0..1....0..0..0..0..0..0....0..1..1..1..1..1
%e ..0..1..0..1..0..1....0..0..0..0..0..1....0..0..0..0..1..1....1..1..1..1..1..1
%e ..1..0..1..0..1..1....0..0..0..0..1..1....0..0..1..1..1..1....1..1..1..1..1..1
%e ..0..1..0..1..1..1....0..0..0..0..1..1....0..0..1..1..1..1....1..1..1..1..1..1
%e ..1..0..1..1..1..1....0..0..0..1..1..1....0..0..1..1..1..1....1..1..1..1..1..1
%e ..0..1..1..1..1..1....0..0..1..1..1..1....0..0..1..1..1..1....1..1..1..1..1..1
%Y Cf. A204651.
%K nonn
%O 1,1
%A _R. H. Hardin_, Jan 17 2012
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