A334281 Number of n-colorings of the vertices of the 4-dimensional cross polytope such that no two adjacent vertices have the same color.

0, 0, 0, 0, 24, 600, 7560, 61320, 351120, 1515024, 5266800, 15531120, 40308840, 94534440, 204228024, 412284600, 786283680, 1428742560, 2490276960, 4186173024, 6816915000, 10793253240, 16666437480, 25164280680, 37233759024, 54090894000, 77278702800

Offset: 0

Author

Peter Kagey, Apr 21 2020

Keywords

• nonn
• easy

Comments

The 4-dimensional cross-polytope is sometimes called the 16-cell. It is one of the six convex regular 4-polytopes.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Eric Weisstein's World of Mathematics, Chromatic Polynomial
• Wikipedia, Cross-polytope
• Wikipedia, Turán graph
• Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Crossrefs

Cf. A091940 (2-dimensional), A115400 (3-dimensional).

Cf. A334279.

Programs

PARI

concat([0,0,0,0], Vec(24*x^4*(1 + 16*x + 126*x^2 + 536*x^3 + 1001*x^4) / (1 - x)^9 + O(x^30))) \\ Colin Barker, Apr 22 2020

Formula

a(n) = n*(n - 1)*(n - 2)*(n - 3)*(465 - 392n + 125n^2 - 18n^3 + n^4).

a(n) = -2790n + 7467n^2 - 7852n^3 + 4300n^4 - 1346n^5 + 244n^6 - 24n^7 + n^8.

From Colin Barker, Apr 22 2020: (Start)

G.f.: 24*x^4*(1 + 16*x + 126*x^2 + 536*x^3 + 1001*x^4) / (1 - x)^9.

a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9) for n>8.

(End)