|
%I #40 Mar 25 2022 14:14:13
%S 0,0,0,0,1,20,120,455,1330,3276,7140,14190,26235,45760,76076,121485,
%T 187460,280840,410040,585276,818805,1125180,1521520,2027795,2667126,
%U 3466100,4455100,5668650,7145775,8930376,11071620,13624345,16649480,20214480
%N a(n) = n*(n-1)*(n-2)*(n-3)*(n^2-3*n-2)/48.
%C 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
%H Solomon W. Golomb, <a href="http://www.jstor.org/stable/2321859">Iterated binomial coefficients</a>, Amer. Math. Monthly, 87 (1980), 719-727.
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F a(n) = binomial(binomial(n-1, 2), 3).
%F G.f.: -x^4*(1+13*x+x^2)/(x-1)^7. - _R. J. Mathar_, Dec 08 2010
%F 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
%F a(n) = A000217(n-1)*A239352(n-2)/6. - _R. J. Mathar_, Mar 25 2022
%e 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
%t Table[ Binomial[ Binomial[n-1, 2], 3], {n,0,32}]
%t LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,0,0,0,1,20,120},40] (* _Harvey P. Dale_, Feb 18 2016 *)
%o (Sage) [(binomial(binomial(n,2),3)) for n in range(-1, 33)] # _Zerinvary Lajos_, Nov 30 2009
%o (PARI) a(n)=n*(n-1)*(n-2)*(n-3)*(n^2-3*n-2)/48 \\ _Charles R Greathouse IV_, Jun 11 2015
%Y From _Robert A. Russell_, Sep 28 2020: (Start)
%Y Cf. A047780 (oriented), A198833 (unoriented), A337898 (achiral) colorings.
%Y a(n+1) = A325006(3,n) (chiral pairs of colorings of orthotope facets or orthoplex vertices.
%Y a(n+1) = A337889(3,n) (chiral pairs of colorings of orthotope faces or orthoplex peaks).
%Y Other polyhedra: A000332 (tetrahedron), A337896 (cube/octahedron).
%Y (End)
%K nonn,easy
%O 0,6
%A _Robert G. Wilson v_ and _Santi Spadaro_, Mar 31 2004
%E Edited (with a new definition) by _N. J. A. Sloane_, Jul 02 2008
|
