%I #21 Aug 01 2023 14:45:00

%S 0,0,0,6,32640,13638780,1034019840,29699591250,460772395776,

%T 4674233282040,34753231503360,203842711924830,991765602960000,

%U 4148317444266996,15316041761879040,50925154505624490,154877550296286720,436185098521110000,1148935457273020416

%N Number of n-colorings of the 5 X 5 X 5 triangular grid.

%C The 5 X 5 X 5 triangular grid has 5 rows with k vertices in row k. Each vertex is connected to the neighbors in the same row and up to two vertices in each of the neighboring rows. The graph has 15 vertices and 30 edges altogether.

%H Alois P. Heinz, <a href="/A182790/b182790.txt">Table of n, a(n) for n = 0..1000</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Chromatic_polynomial">Chromatic polynomial</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_graph#Other_kinds">Triangular grid graph</a>

%H <a href="/index/Rec#order_16">Index entries for linear recurrences with constant coefficients</a>, signature (16,-120,560,-1820,4368,-8008,11440,-12870,11440,-8008,4368,-1820,560,-120,16,-1).

%F a(n) = n*(n-1)*(n^9 -21*n^8 +198*n^7 -1102*n^6 +3999*n^5 -9840*n^4 +16475*n^3 -18177*n^2 +12056*n -3686)*(n-2)^4.

%F G.f.: 6*x^3*(1769985*x^12 +130265584*x^11 +2438678946*x^10 +17020599920*x^9 +51993920175*x^8 +74836435680*x^7 +51909140892*x^6 +17013829728*x^5 +2462276655*x^4 +136618800*x^3 +2186210*x^2 +5424*x +1)/(x-1)^16.

%p a:= n-> n^15 -30*n^14 +419*n^13 -3612*n^12 +21477*n^11 -93207*n^10 +304555*n^9 -761340*n^8 +1463473*n^7 -2152758*n^6 +2385118*n^5 -1929184*n^4 +1075936*n^3 -369824*n^2 +58976*n:

%p seq(a(n), n=0..30);

%Y Column k=5 of A182797.

%Y Cf. A178435, A182798, A182788, A182789, A182791, A182792, A182793, A182794, A182795, A182796.

%K nonn,easy

%O 0,4

%A _Alois P. Heinz_, Dec 02 2010