A338959 Number of achiral colorings of the 96 edges (or triangular faces) of the 4-D 24-cell using exactly n colors.

1, 24124751133507582, 883287060135783817036973460, 27692672230411020835164184856095160, 18069944152044184972628509749308321354400, 1018093811663859334508633754250963606821400320

Offset: 1

Author

Robert A. Russell, Nov 17 2020

Keywords

• fini
• nonn
• full

Comments

An achiral coloring is identical to its reflection. The Schläfli symbol of the 24-cell is {3,4,3}. It has 24 octahedral facets. It is self-dual. For n>60, a(n) = 0.

Links

• Robert A. Russell, Table of n, a(n) for n = 1..60

Crossrefs

Cf. A338956 (oriented), A338957 (unoriented), A338958 (chiral), A338955 (up to n colors), A338951 (vertices, facets), A331353 (5-cell), A331361 (8-cell edges, 16-cell faces), A331357 (16-cell edges, 8-cell faces), A338983 (120-cell, 600-cell).

Programs

Mathematica

bp[j_] := Sum[k! StirlingS2[j, k] x^k, {k, 0, j}] (*binomial series*)
Drop[CoefficientList[bp[16]/6+bp[18]/6+bp[20]/3+bp[24]/4+bp[48]/24+bp[52]/48+bp[60]/48,x],1]

Formula

A338955(n) = Sum_{j=1..Min(n,60)} a(n) * binomial(n,j).

a(n) = 2*A338957(n) - A338956(n) = A338956(n) - 2*A338958(n) = A338957(n) - A338958(n).