A093566 a(n) = n*(n-1)*(n-2)*(n-3)*(n^2-3*n-2)/48.

0, 0, 0, 0, 1, 20, 120, 455, 1330, 3276, 7140, 14190, 26235, 45760, 76076, 121485, 187460, 280840, 410040, 585276, 818805, 1125180, 1521520, 2027795, 2667126, 3466100, 4455100, 5668650, 7145775, 8930376, 11071620, 13624345, 16649480, 20214480

Offset

0,6

Comments

a(n+1) is the number of chiral pairs of colorings of the faces of a cube (vertices of a regular octahedron) using n or fewer colors. - Robert A. Russell, Sep 28 2020

Links

Table of n, a(n) for n=0..33.

Solomon W. Golomb, Iterated binomial coefficients, Amer. Math. Monthly, 87 (1980), 719-727.

Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Formula

a(n) = binomial(binomial(n-1, 2), 3).

G.f.: -x^4*(1+13*x+x^2)/(x-1)^7. - R. J. Mathar, Dec 08 2010

a(n+1) = 1*C(n,3) + 16*C(n,4) + 30*C(n,5) + 15*C(n,6), where the coefficient of C(n,k) is the number of chiral pairs of colorings using exactly k colors. - Robert A. Russell, Sep 28 2020

a(n) = A000217(n-1)*A239352(n-2)/6. - R. J. Mathar, Mar 25 2022

Example

For a(3+1) = 1, each of the three colors is applied to a pair of adjacent faces of the cube (vertices of the octahedron). - Robert A. Russell, Sep 28 2020

Mathematica

Table[ Binomial[ Binomial[n-1, 2], 3], {n, 0, 32}]
LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {0, 0, 0, 0, 1, 20, 120}, 40] (* Harvey P. Dale, Feb 18 2016 *)

Prog

(Sage) [(binomial(binomial(n, 2), 3)) for n in range(-1, 33)] # Zerinvary Lajos, Nov 30 2009
(PARI) a(n)=n*(n-1)*(n-2)*(n-3)*(n^2-3*n-2)/48 \\ Charles R Greathouse IV, Jun 11 2015

Crossrefs

From Robert A. Russell, Sep 28 2020: (Start)

Cf. A047780 (oriented), A198833 (unoriented), A337898 (achiral) colorings.

a(n+1) = A325006(3,n) (chiral pairs of colorings of orthotope facets or orthoplex vertices.

a(n+1) = A337889(3,n) (chiral pairs of colorings of orthotope faces or orthoplex peaks).

Other polyhedra: A000332 (tetrahedron), A337896 (cube/octahedron).

(End)

Sequence in context: A044352 A044733 A280439 * A213223 A041770 A156255

Adjacent sequences: A093563 A093564 A093565 * A093567 A093568 A093569

Keyword

nonn,easy

Author

Robert G. Wilson v and Santi Spadaro, Mar 31 2004

Extensions

Edited (with a new definition) by N. J. A. Sloane, Jul 02 2008

Status

approved