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 A331350 #17 Mar 10 2024 13:19:13
%S A331350 1,40,1197,18592,166885,1019880,4738153,17962624,58248153,166920040,
%T A331350 432738229,1032709536,2298857821,4822806184,9613704465,18329410048,
%U A331350 33605960689,59516325288,102196242685,170682720160,278019522837
%N A331350 Number of oriented colorings of the edges (or triangular faces) of a regular 4-dimensional simplex with n available colors.
%C A331350 A 4-dimensional simplex has 5 vertices and 10 edges. Its Schläfli symbol is {3,3,3}. Two oriented colorings are the same if one is a rotation of the other; chiral pairs are counted as two.
%C A331350 There are 60 elements in the rotation group of the 4-dimensional simplex. Each is an even permutation of the vertices and can be associated with a partition of 5 based on the conjugacy group of the permutation. The first formula is obtained by averaging their cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.
%C A331350 Partition Count Even Cycle Indices
%C A331350 5 24 x_5^2
%C A331350 311 20 x_1^1x_3^3
%C A331350 221 15 x_1^2x_2^4
%C A331350 11111 1 x_1^10
%H A331350 G. Royle, <a href="http://teaching.csse.uwa.edu.au/units/CITS7209/partition.pdf">Partitions and Permutations</a>
%H A331350 <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1).
%F A331350 a(n) = (24*n^2 + 20*n^4 + 15*n^6 + n^10) / 60.
%F A331350 a(n) = C(n,1) + 38*C(n,2) + 1080*C(n,3) + 14040*C(n,4) + 85500*C(n,5) + 274104*C(n,6) + 493920*C(n,7) + 504000*C(n,8) + 272160*C(n,9) + 60480*C(n,10), where the coefficient of C(n,k) is the number of colorings using exactly k colors.
%F A331350 a(n) = A063843(n) + A331352(n) = 2*A063843(n) - A331353(n) = 2*A331352(n) + A331353(n).
%t A331350 Table[(24n^2 + 20n^4 + 15n^6 + n^10)/60, {n, 1, 25}]
%Y A331350 Cf. A063843 (unoriented), A331352 (chiral), A331353 (achiral).
%Y A331350 Other polychora: A331358 (8-cell), A331354 (16-cell), A338952 (24-cell), A338964 (120-cell, 600-cell).
%Y A331350 Row 4 of A327083 (simplex edges and facets) and A337883 (simplex faces and peaks).
%K A331350 nonn,easy
%O A331350 1,2
%A A331350 _Robert A. Russell_, Jan 14 2020