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A250855 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.

%I A250855 #7 Nov 21 2018 16:12:16
%S A250855 2670,13097,44797,123016,290646,614965,1195457,2172712,3738406,
%T A250855 6146361,9724685,14888992,22156702,32162421,45674401,63612080,
%U A250855 87064702,117311017,155840061,204373016,264886150,339634837,431178657,542407576,676569206
%N A250855 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 A250855 R. H. Hardin, <a href="/A250855/b250855.txt">Table of n, a(n) for n = 1..210</a>
%F A250855 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 A250855 Conjectures from _Colin Barker_, Nov 21 2018: (Start)
%F A250855 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 A250855 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 A250855 (End)
%e A250855 Some solutions for n=4:
%e A250855 ..3..3..3..1..1....2..2..2..2..3....3..3..2..2..2....1..2..1..1..1
%e A250855 ..1..1..1..1..2....1..1..1..1..2....1..1..1..1..1....0..1..1..1..1
%e A250855 ..1..1..1..1..2....1..1..1..2..3....1..1..2..2..2....0..1..2..2..2
%e A250855 ..0..0..0..1..2....0..0..0..1..3....0..0..2..3..3....0..1..2..3..3
%Y A250855 Row 3 of A250853.
%K A250855 nonn
%O A250855 1,1
%A A250855 _R. H. Hardin_, Nov 28 2014