This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A337952 #15 Mar 10 2024 13:24:20
%S A337952 1,496,230076,22456756,795467350,14697611496,173107727191,
%T A337952 1466088119056,9651378868011,52083991149400,239323201136866,
%U A337952 962942859342036,3465720389989936,11343525530430016,34210497067620525
%N A337952 Number of oriented colorings of the 16 tetrahedral facets of a hyperoctahedron or of the 16 vertices of a tesseract.
%C A337952 Each chiral pair is counted as two when enumerating oriented arrangements. 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. There are 192 elements in the rotation group of the tesseract. Each involves a permutation of the axes that can be associated with a partition of 4 based on the conjugacy class of the permutation. This table shows the cycle indices for each rotation by partition. The first formula is obtained by averaging these cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.
%C A337952 Partition Count Even Cycle Indices
%C A337952 4 6 8x_8^2
%C A337952 31 8 4x_1^4x_3^4 + 4x_2^2x_6^2
%C A337952 22 3 4x_1^4x_2^6 + 4x_4^4
%C A337952 211 6 4x_2^8 + 4x_4^4
%C A337952 1111 1 x_1^16 + 7x_2^8
%H A337952 <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 A337952 a(n) = n^2 * (n^14 + 12*n^8 + 63*n^6 + 68*n^2 + 48) / 192.
%F A337952 a(n) = 1*C(n,1) + 494*C(n,2) + 228591*C(n,3) + 21539424*C(n,4) + 685479375*C(n,5) + 10257064650*C(n,6) + 86151316860*C(n,7) + 449772354360*C(n,8) + 1551283253100*C(n,9) + 3661969537800*C(n,10) + 6015983173200*C(n,11) + 6878457986400*C(n,12) + 5371454088000*C(n,13) + 2733402672000*C(n,14) + 817296480000*C(n,15) + 108972864000*C(n,16), where the coefficient of C(n,k) is the number of oriented colorings using exactly k colors.
%F A337952 a(n) = A128767(n) + A337954(n) = 2*A128767(n) - A337955(n) = 2*A337954(n) + A337955(n).
%t A337952 Table[(n^16+12n^10+63n^8+68n^4+48n^2)/192,{n,30}]
%Y A337952 Cf. A128767 (unoriented), A337954 (chiral), A337955 (achiral).
%Y A337952 Other elements: A331358 (tesseract edges, hyperoctahedron faces), A331354 (tesseract faces, hyperoctahedron edges), A337956 (tesseract facets, hyperoctahedron vertices).
%Y A337952 Other polychora: A337895 (4-simplex facets/vertices), A338948 (24-cell), A338964 (120-cell, 600-cell).
%Y A337952 Row 4 of A325012 (orthoplex facets, orthotope vertices).
%K A337952 nonn,easy
%O A337952 1,2
%A A337952 _Robert A. Russell_, Oct 03 2020