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A182790 Number of n-colorings of the 5 X 5 X 5 triangular grid.

%I A182790 #21 Aug 01 2023 14:45:00
%S A182790 0,0,0,6,32640,13638780,1034019840,29699591250,460772395776,
%T A182790 4674233282040,34753231503360,203842711924830,991765602960000,
%U A182790 4148317444266996,15316041761879040,50925154505624490,154877550296286720,436185098521110000,1148935457273020416
%N A182790 Number of n-colorings of the 5 X 5 X 5 triangular grid.
%C A182790 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 A182790 Alois P. Heinz, <a href="/A182790/b182790.txt">Table of n, a(n) for n = 0..1000</a>
%H A182790 Wikipedia, <a href="https://en.wikipedia.org/wiki/Chromatic_polynomial">Chromatic polynomial</a>
%H A182790 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_graph#Other_kinds">Triangular grid graph</a>
%H A182790 <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 A182790 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 A182790 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 A182790 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 A182790 seq(a(n), n=0..30);
%Y A182790 Column k=5 of A182797.
%Y A182790 Cf. A178435, A182798, A182788, A182789, A182791, A182792, A182793, A182794, A182795, A182796.
%K A182790 nonn,easy
%O A182790 0,4
%A A182790 _Alois P. Heinz_, Dec 02 2010