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

0,5

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).

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)

Prog

(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

Crossrefs

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

Cf. A334279.

Sequence in context: A042106 A158637 A179059 * A055359 A154128 A362513

Adjacent sequences: A334278 A334279 A334280 * A334282 A334283 A334284

Keyword

nonn,easy

Author

Peter Kagey, Apr 21 2020

Status

approved