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

0, 0, 0, 6, 2112, 98820, 1574400, 13676250, 80631936, 363204072, 1342218240, 4261697550, 12000120000, 30653510316, 72237215232, 159067919010, 330577363200, 653537970000, 1236951760896, 2253171240342, 3967187906880, 6776444390100, 11264003520000, 18268445544426

Offset: 0

Author

Alois P. Heinz, Dec 02 2010

Keywords

• nonn
• easy

Comments

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.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Wikipedia, Chromatic polynomial
• Wikipedia, Triangular grid graph
• Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Crossrefs

Column k=4 of A182797.

Cf. A178435, A182798, A182788, A182790, A182791, A182792, A182793, A182794, A182795, A182796.

Programs

Maple

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:
seq(a(n), n=0..30);

Formula

a(n) = n*(n-1)*(n-2)^4*(n^4-9*n^3+31*n^2-49*n+31).

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