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A182792 Number of n-colorings of the 7 X 7 X 7 triangular grid.

%I A182792 #20 Jan 21 2024 11:53:54
%S A182792 0,0,0,6,22665216,3013780700340,14668353488732160,
%T A182792 10142843465859326250,2047585083571827265536,177922459478738381782536,
%U A182792 8379024937788205785415680,247855358008532074713890670,5089690042011600555045120000
%N A182792 Number of n-colorings of the 7 X 7 X 7 triangular grid.
%C A182792 The 7 X 7 X 7 triangular grid has 7 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 28 vertices and 63 edges altogether.
%H A182792 Alois P. Heinz, <a href="/A182792/b182792.txt">Table of n, a(n) for n = 0..1000</a>
%H A182792 Wikipedia, <a href="https://en.wikipedia.org/wiki/Chromatic_polynomial">Chromatic polynomial</a>
%H A182792 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_graph#Other_kinds">Triangular grid graph</a>
%H A182792 <a href="/index/Rec#order_29">Index entries for linear recurrences with constant coefficients</a>, signature (29, -406, 3654, -23751, 118755, -475020, 1560780, -4292145, 10015005, -20030010, 34597290, -51895935, 67863915, -77558760, 77558760, -67863915, 51895935, -34597290, 20030010, -10015005, 4292145, -1560780, 475020, -118755, 23751, -3654, 406, -29, 1).
%F A182792 a(n) = n^28 -63*n^27 + ... (see Maple program).
%p A182792 a:= n-> n^28 -63*n^27 +1917*n^26 -37515*n^25 +530415*n^24 -5770367*n^23 +50229001*n^22 -359161581*n^21 +2149154827*n^20 -10907730408*n^19 +47421892719*n^18 -177879720291*n^17 +578605822984*n^16 -1637514420168*n^15 +4039146735437*n^14 -8685828617404*n^13 +16263806276269*n^12 -26442885506316*n^11 +37160408393135*n^10 -44834641208000*n^9 +46004737388704*n^8 -39623922744400*n^7
%p A182792 +28128173982016*n^6 -16032767219808*n^5 +7056432341248*n^4 -2251430597120*n^3 +463382611200*n^2 -46172626432*n: seq(a(n), n=0..30);
%Y A182792 7th column of A182797. Cf. A178435, A182798, A182788, A182789, A182790, A182791, A182793, A182794, A182795, A182796.
%K A182792 nonn,easy
%O A182792 0,4
%A A182792 _Alois P. Heinz_, Dec 02 2010