A338957 Number of unoriented colorings of the 96 edges (or triangular faces) of the 4-D 24-cell using exactly n colors.

1, 68774446639102959610154174, 5523164445430505754875774375105924818979901, 5448873034167734394172913824852272971748608894646534804, 10956401434158576570935668826433407535831446552957081921713485225

Offset: 1

Author

Robert A. Russell, Nov 17 2020

Keywords

• fini
• nonn
• full

Comments

Each chiral pair is counted as one when enumerating unoriented arrangements. 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.

Links

• Robert A. Russell, Table of n, a(n) for n = 1..96

Crossrefs

Cf. A338956 (oriented), A338958 (chiral), A338959 (achiral), A338953 (up to n colors), A338949 (vertices, facets), A063843 (5-cell), A331359 (8-cell edges, 16-cell faces), A331355 (16-cell edges, 8-cell faces), A338981 (120-cell, 600-cell).

Programs

Mathematica

bp[j_] := Sum[k! StirlingS2[j, k] x^k, {k, 0, j}] (* binomial series *)
Drop[CoefficientList[bp[8]/12+bp[12]/8+bp[16]/8+bp[18]/9+bp[20]/6+19bp[24]/96+bp[32]/24+bp[36]/36+43bp[48]/1152+bp[50]/16+bp[52]/96+bp[60]/96+bp[96]/1152,x],1]

Formula

A338953(n) = Sum_{j=1..Min(n,96)} a(n) * binomial(n,j).

a(n) = A338956(n) - A338958(n) = (A338956(n) + A338959(n)) / 2 = A338958(n) + A338959(n).