%I #13 Mar 09 2024 12:13:06

%S 0,94,97974,10700090,390081800,7280687610,86121007714,730895668104,

%T 4816861200630,26010740238450,119563513291420,481192778757834,

%U 1732132086737234,5669991002636870,17101193825828700,48029634770843680

%N Number of chiral pairs of colorings of the 16 tetrahedral facets of a hyperoctahedron or of the 16 vertices of a tesseract.

%C Each member of a chiral pair is a reflection, but not a rotation, of the other. The Schläfli symbols for the tesseract and the hyperoctahedron are {4,3,3} and {3,3,4} respectively. Both figures are regular 4-D polyhedra and they are mutually dual.

%H <a href="/index/Rec#order_17">Index entries for linear recurrences with constant coefficients</a>, signature (17, -136, 680, -2380, 6188, -12376, 19448, -24310, 24310, -19448, 12376, -6188, 2380, -680, 136, -17, 1).

%F a(n) = (n-1) * n^2 * (n+1) * (n^12 + n^10 - 11*n^8 + n^6 + 44 n^4 - 4 n^2 - 48) / 384.

%F a(n) = 94*C(n,2) + 97692*C(n,3) + 10308758*C(n,4) + 337560150*C(n,5) + 5098740090*C(n,6) + 42976836210*C(n,7) + 224685801060*C(n,8) + 775389028050*C(n,9) + 1830791421900*C(n,10) + 3007909258200*C(n,11) + 3439214024400*C(n,12) + 2685727044000*C(n,13) + 1366701336000*C(n,14) + 408648240000*C(n,15) + 54486432000*C(n,16), where the coefficient of C(n,k) is the number of chiral pairs of colorings using exactly k colors.

%F a(n) = A337952(n) - A128767(n) = (A337952(n) - A337955(n)) / 2 = A128767(n) - A337955(n).

%t Table[(n^16-12n^12+12n^10+43n^8-48n^6-44n^4+48n^2)/384,{n, 30}]

%Y Cf. A337952 (oriented), A128767 (unoriented), A337955 (achiral).

%Y Other elements: A331360 (tesseract edges, hyperoctahedron faces), A331356 (tesseract faces, hyperoctahedron edges), A234249(n+1) (tesseract facets, hyperoctahedron vertices).

%Y Other polychora: A000389 (4-simplex facets/vertices), A338950 (24-cell), A338966 (120-cell, 600-cell).

%Y Row 4 of A325014 (orthoplex facets, orthotope vertices).

%K nonn,easy

%O 1,2

%A _Robert A. Russell_, Oct 03 2020