A338958 - OEIS

%I #14 Dec 20 2020 02:21:50

%S 68774446614978208476646592,

%T 5523164445430504871588714239322107782006441,

%U 5448873034167734394145221152621861950913444709790439644,10956401434158576570935650756489255491646473924447332613392130825

%N Number of chiral pairs of colorings of the 96 edges (or triangular faces) of the 4-D 24-cell using exactly n colors.

%C Each member of a chiral pair is a reflection but not a rotation of the other. The Schläfli symbol of the 24-cell is {3,4,3}. It has 24 octahedral facets. It is self-dual. For n>96, a(n) = 0.

%H Robert A. Russell, <a href="/A338958/b338958.txt">Table of n, a(n) for n = 2..96</a>

%F A338954(n) = Sum_{j=2..Min(n,96)} a(n) * binomial(n,j).

%F a(n) = A338956(n) - A338957(n) = (A338956(n) - A338959(n)) / 2 = A338957(n) - A338959(n).

%t bp[j_] := Sum[k! StirlingS2[j, k] x^k, {k, 0, j}] (*binomial series*)

%t Drop[CoefficientList[bp[8]/12+bp[12]/8-bp[16]/24-bp[18]/18-bp[20]/6-5bp[24]/96+bp[32]/24+bp[36]/36-5bp[48]/1152+bp[50]/16-bp[52]/96-bp[60]/96+bp[96]/1152,x],2]

%Y Cf. A338956 (oriented), A338957 (unoriented), A338959 (achiral), A338954 (up to n colors), A338950 (vertices, facets), A331352 (5-cell), A331360 (8-cell edges, 16-cell faces), A331356 (16-cell edges, 8-cell faces), A338982 (120-cell, 600-cell).

%K fini,nonn,full

%O 2,1

%A _Robert A. Russell_, Nov 17 2020