%I #8 Jun 02 2018 10:35:18
%S 1,10,90,534,2310,8012,23661,61830,146718,321970,662233,1289652,
%T 2396745,4277352,7367630,12299364,19968183,31619610,48956236,74269690,
%U 110601480,161937204,233439075,331722170,465180300,644367906,882444915,1195692040
%N Number of n X 3 nonnegative integer arrays with each row and column increasing from zero by 0, 1, 2 or 3.
%C Column 3 of A202924.
%H R. H. Hardin, <a href="/A202919/b202919.txt">Table of n, a(n) for n = 1..210</a>
%F Empirical: a(n) = (1/17280)*n^9 + (23/20160)*n^8 + (193/20160)*n^7 + (13/288)*n^6 + (523/5760)*n^5 + (1/2880)*n^4 + (29/4320)*n^3 + (457/1008)*n^2 + (11/28)*n.
%F Conjectures from _Colin Barker_, Jun 02 2018: (Start)
%F G.f.: x*(1 + 35*x^2 - 36*x^3 + 30*x^4 - 10*x^5 + x^6) / (1 - x)^10.
%F 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>10.
%F (End)
%e Some solutions for n=5:
%e ..0..0..0....0..0..0....0..0..0....0..0..0....0..0..0....0..0..0....0..0..0
%e ..0..2..2....0..1..1....0..0..1....0..0..2....0..0..2....0..0..2....0..0..1
%e ..0..2..3....0..1..1....0..2..2....0..1..4....0..1..2....0..1..2....0..0..2
%e ..0..3..6....0..3..3....0..3..5....0..3..5....0..1..2....0..1..2....0..0..2
%e ..0..3..6....0..3..4....0..3..6....0..3..5....0..2..2....0..2..5....0..2..5
%Y Cf. A202924.
%K nonn
%O 1,2
%A _R. H. Hardin_, Dec 26 2011