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A278772 Number of n X 2 0..1 arrays with rows in nondecreasing lexicographic order and columns in nonincreasing lexicographic order, but with exactly two mistakes.

%I A278772 #7 Feb 10 2019 08:54:30
%S A278772 0,2,20,117,503,1750,5209,13751,33000,73282,152581,300872,566293,
%T A278772 1023724,1786462,3021818,4971616,7978746,12521114,19254543,29066411,
%U A278772 43142066,63046335,90822745,129113400,181302810,251689347,345688410,470071817
%N A278772 Number of n X 2 0..1 arrays with rows in nondecreasing lexicographic order and columns in nonincreasing lexicographic order, but with exactly two mistakes.
%H A278772 R. H. Hardin, <a href="/A278772/b278772.txt">Table of n, a(n) for n = 1..210</a>
%F A278772 Empirical: a(n) = (1/3628800)*n^10 + (1/80640)*n^9 + (1/3780)*n^8 + (17/8064)*n^7 + (1513/172800)*n^6 + (167/11520)*n^5 + (12329/362880)*n^4 + (197/4032)*n^3 - (2167/50400)*n^2 - (11/168)*n.
%F A278772 Conjectures from _Colin Barker_, Feb 10 2019: (Start)
%F A278772 G.f.: x^2*(2 - 2*x + 7*x^2 - 14*x^3 + 12*x^4 - 5*x^5 + x^6) / (1 - x)^11.
%F A278772 a(n) = 11*a(n-1) - 55*a(n-2) + 165*a(n-3) - 330*a(n-4) + 462*a(n-5) - 462*a(n-6) + 330*a(n-7) - 165*a(n-8) + 55*a(n-9) - 11*a(n-10) + a(n-11) for n>11.
%F A278772 (End)
%e A278772 Some solutions for n=4:
%e A278772 ..0..0. .0..1. .1..1. .1..1. .1..1. .0..1. .0..1. .0..1. .1..1. .1..0
%e A278772 ..0..1. .0..0. .1..0. .1..0. .1..0. .0..0. .1..0. .1..0. .1..1. .0..1
%e A278772 ..0..0. .1..0. .1..0. .0..1. .0..0. .0..1. .1..1. .1..1. .1..0. .1..0
%e A278772 ..0..1. .1..0. .0..0. .0..1. .1..1. .1..1. .1..0. .0..0. .0..1. .0..0
%Y A278772 Column 2 of A278778.
%K A278772 nonn
%O A278772 1,2
%A A278772 _R. H. Hardin_, Nov 28 2016