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Number of nX3 nonnegative integer arrays with upper left 0 and lower right n+3-5 and value increasing by 0 or 1 with every step right or down.
1

%I #9 Nov 23 2015 05:53:12

%S 0,1,18,153,770,2859,8694,22924,54272,118057,239798,460207,841930,

%T 1478451,2505634,4116442,6579440,10261761,15657290,23420901,34409666,

%U 49732043,70806142,99428264,137853008,188886345,255993166,343420923

%N Number of nX3 nonnegative integer arrays with upper left 0 and lower right n+3-5 and value increasing by 0 or 1 with every step right or down.

%C Column 3 of A252976.

%H R. H. Hardin, <a href="/A252971/b252971.txt">Table of n, a(n) for n = 1..210</a>

%F Empirical: a(n) = (1/90720)*n^9 + (1/2520)*n^8 + (19/3780)*n^7 + (7/360)*n^6 - (59/4320)*n^5 - (19/180)*n^4 + (5827/45360)*n^3 - (32/35)*n^2 + (3629/1260)*n - 2.

%F Empirical: G.f.: x^2*(1+8*x+18*x^2-70*x^3+94*x^4-78*x^5+43*x^6-14*x^7+2*x^8) / (x-1)^10. - _R. J. Mathar_, Nov 23 2015

%e Some solutions for n=4

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

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

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

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

%K nonn

%O 1,3

%A _R. H. Hardin_, Dec 25 2014