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A252817 Number of n X 5 nonnegative integer arrays with upper left 0 and every value within 2 of its city block distance from the upper left and every value increasing by 0 or 1 with every step right or down. 1

%I #10 Dec 06 2018 10:10:26

%S 11,81,468,2078,7564,23664,65711,165685,385736,839799,1726761,3379640,

%T 6336411,11439478,19972358,33843927,55832593,89905021,141626554,

%U 218683266,331538662,494251420,725484265,1049738089,1498849798

%N Number of n X 5 nonnegative integer arrays with upper left 0 and every value within 2 of its city block distance from the upper left and every value increasing by 0 or 1 with every step right or down.

%H R. H. Hardin, <a href="/A252817/b252817.txt">Table of n, a(n) for n = 1..210</a>

%F Empirical: a(n) = (1/259200)*n^10 + (1/6480)*n^9 + (149/60480)*n^8 + (163/7560)*n^7 + (10411/86400)*n^6 + (209/432)*n^5 + (17977/12960)*n^4 + (6043/3240)*n^3 + (37673/12600)*n^2 + (1423/1260)*n + 3.

%F Conjectures from _Colin Barker_, Dec 06 2018: (Start)

%F G.f.: x*(11 - 40*x + 182*x^2 - 430*x^3 + 711*x^4 - 822*x^5 + 657*x^6 - 360*x^7 + 131*x^8 - 29*x^9 + 3*x^10) / (1 - x)^11.

%F a(n) = 11*a(n-1) - 55*a(n-2) + 165*a(n-3) - 330*a(n-4) + 462*a(n-5) - 462*a(n-6) + 330*a(n-7) - 165*a(n-8) + 55*a(n-9) - 11*a(n-10) + a(n-11) for n>11.

%F (End)

%e Some solutions for n=4:

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

%e ..1..2..2..3..3....1..2..2..3..4....1..1..2..3..4....1..1..2..3..4

%e ..2..3..3..4..4....2..2..2..3..4....1..2..3..4..4....1..2..2..3..4

%e ..2..3..4..5..5....2..2..3..4..5....2..3..3..4..5....1..2..3..4..5

%Y Column 5 of A252820.

%K nonn

%O 1,1

%A _R. H. Hardin_, Dec 22 2014

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