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Number of (3+1) X (n+1) 0..3 arrays with nondecreasing x(i,j)-x(i,j-1) in the i direction and nondecreasing min(x(i,j),x(i-1,j)) in the j direction.
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%I #7 Nov 21 2018 16:12:16

%S 2670,13097,44797,123016,290646,614965,1195457,2172712,3738406,

%T 6146361,9724685,14888992,22156702,32162421,45674401,63612080,

%U 87064702,117311017,155840061,204373016,264886150,339634837,431178657,542407576,676569206

%N Number of (3+1) X (n+1) 0..3 arrays with nondecreasing x(i,j)-x(i,j-1) in the i direction and nondecreasing min(x(i,j),x(i-1,j)) in the j direction.

%H R. H. Hardin, <a href="/A250855/b250855.txt">Table of n, a(n) for n = 1..210</a>

%F Empirical: a(n) = (3/2)*n^6 + (74/3)*n^5 + (621/4)*n^4 + (3161/6)*n^3 + (3691/4)*n^2 + 783*n + 256.

%F Conjectures from _Colin Barker_, Nov 21 2018: (Start)

%F G.f.: x*(2670 - 5593*x + 9188*x^2 - 8976*x^3 + 5326*x^4 - 1791*x^5 + 256*x^6) / (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>7.

%F (End)

%e Some solutions for n=4:

%e ..3..3..3..1..1....2..2..2..2..3....3..3..2..2..2....1..2..1..1..1

%e ..1..1..1..1..2....1..1..1..1..2....1..1..1..1..1....0..1..1..1..1

%e ..1..1..1..1..2....1..1..1..2..3....1..1..2..2..2....0..1..2..2..2

%e ..0..0..0..1..2....0..0..0..1..3....0..0..2..3..3....0..1..2..3..3

%Y Row 3 of A250853.

%K nonn

%O 1,1

%A _R. H. Hardin_, Nov 28 2014