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A219596 Number of 3 X n arrays of the minimum value of corresponding elements and their horizontal or vertical neighbors in a random, but sorted with lexicographically nondecreasing rows and nonincreasing columns, 0..2 3 X n array.

%I A219596 #9 Jul 23 2025 00:34:04
%S A219596 6,21,84,233,550,1188,2415,4684,8746,15833,27945,48286,81907,136629,
%T A219596 224336,362747,577797,906780,1402432,2138159,3214644,4768098,6980453,
%U A219596 10091830,14415652,20356811,28433339,39302076,53788873,72923915,97982798
%N A219596 Number of 3 X n arrays of the minimum value of corresponding elements and their horizontal or vertical neighbors in a random, but sorted with lexicographically nondecreasing rows and nonincreasing columns, 0..2 3 X n array.
%C A219596 Row 3 of A219595.
%H A219596 R. H. Hardin, <a href="/A219596/b219596.txt">Table of n, a(n) for n = 1..210</a>
%F A219596 Empirical: a(n) = (1/181440)*n^9 - (1/8064)*n^8 + (11/3780)*n^7 - (107/2880)*n^6 + (749/1728)*n^5 - (1943/1152)*n^4 + (168241/90720)*n^3 + (355057/10080)*n^2 - (306913/2520)*n + 137 for n>3.
%F A219596 Conjectures from _Colin Barker_, Jul 26 2018: (Start)
%F A219596 G.f.: x*(6 - 39*x + 144*x^2 - 382*x^3 + 740*x^4 - 1009*x^5 + 933*x^6 - 554*x^7 + 195*x^8 - 32*x^9 - 5*x^10 + 7*x^11 - 2*x^12) / (1 - x)^10.
%F A219596 a(n) = 10*a(n-1) - 45*a(n-2) + 120*a(n-3) - 210*a(n-4) + 252*a(n-5) - 210*a(n-6) + 120*a(n-7) - 45*a(n-8) + 10*a(n-9) - a(n-10) for n>13.
%F A219596 (End)
%e A219596 Some solutions for n=3:
%e A219596 ..1..0..0....0..0..0....0..0..0....0..0..0....0..0..0....0..0..0....1..1..1
%e A219596 ..1..0..0....0..0..0....1..0..0....0..0..0....1..0..0....0..0..0....1..1..1
%e A219596 ..2..1..0....2..2..1....0..0..0....1..1..1....1..0..0....1..1..0....2..2..2
%Y A219596 Cf. A219595.
%K A219596 nonn
%O A219596 1,1
%A A219596 _R. H. Hardin_, Nov 23 2012