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Number of n X 3 0..1 arrays with rows, columns and antidiagonals unimodal and diagonals nondecreasing.
1

%I #8 Aug 22 2018 18:04:40

%S 7,28,71,155,317,607,1097,1887,3112,4950,7631,11447,16763,24029,33793,

%T 46715,63582,85324,113031,147971,191609,245627,311945,392743,490484,

%U 607938,748207,914751,1111415,1342457,1612577,1926947,2291242,2711672

%N Number of n X 3 0..1 arrays with rows, columns and antidiagonals unimodal and diagonals nondecreasing.

%C Column 3 of A223770.

%H R. H. Hardin, <a href="/A223765/b223765.txt">Table of n, a(n) for n = 1..210</a>

%F Empirical: a(n) = (1/720)*n^6 + (1/240)*n^5 + (41/144)*n^4 - (13/48)*n^3 + (2237/360)*n^2 - (217/30)*n + 19 for n>2.

%F Conjectures from _Colin Barker_, Aug 22 2018: (Start)

%F G.f.: x*(7 - 21*x + 22*x^2 + x^3 - 12*x^4 - 9*x^5 + 26*x^6 - 17*x^7 + 4*x^8) / (1 - x)^7.

%F a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n>9.

%F (End)

%e Some solutions for n=3:

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

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

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

%Y Cf. A223770.

%K nonn

%O 1,1

%A _R. H. Hardin_, Mar 27 2013