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A182789 Number of n-colorings of the 4 X 4 X 4 triangular grid.

%I A182789 #20 Aug 01 2023 14:35:57
%S A182789 0,0,0,6,2112,98820,1574400,13676250,80631936,363204072,1342218240,
%T A182789 4261697550,12000120000,30653510316,72237215232,159067919010,
%U A182789 330577363200,653537970000,1236951760896,2253171240342,3967187906880,6776444390100,11264003520000,18268445544426
%N A182789 Number of n-colorings of the 4 X 4 X 4 triangular grid.
%C A182789 The 4 X 4 X 4 triangular grid has 4 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 10 vertices and 18 edges altogether.
%H A182789 Alois P. Heinz, <a href="/A182789/b182789.txt">Table of n, a(n) for n = 0..1000</a>
%H A182789 Wikipedia, <a href="https://en.wikipedia.org/wiki/Chromatic_polynomial">Chromatic polynomial</a>
%H A182789 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_graph#Other_kinds">Triangular grid graph</a>
%H A182789 <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).
%F A182789 a(n) = n*(n-1)*(n-2)^4*(n^4-9*n^3+31*n^2-49*n+31).
%F A182789 G.f.: -6*x^3*(3267*x^7 +51359*x^6 +195679*x^5 +241075*x^4 +100425*x^3 +12653*x^2 +341*x +1) / (x -1)^11. - _Colin Barker_, Oct 01 2014
%p A182789 a:= n-> n^10 -18*n^9 +144*n^8 -672*n^7 +2016*n^6 -4031*n^5 +5368*n^4 -4584*n^3 +2272*n^2 -496*n:
%p A182789 seq(a(n), n=0..30);
%Y A182789 Column k=4 of A182797.
%Y A182789 Cf. A178435, A182798, A182788, A182790, A182791, A182792, A182793, A182794, A182795, A182796.
%K A182789 nonn,easy
%O A182789 0,4
%A A182789 _Alois P. Heinz_, Dec 02 2010