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 A128767 #27 Dec 18 2022 05:40:43
%S A128767 1,402,132102,11756666,405385550,7416923886,86986719477,735192450952,
%T A128767 4834517667381,26073250910950,119759687845446,481750080584202,
%U A128767 1733588303252702,5673534527793146,17109303241791825,48047227408513056
%N A128767 Number of inequivalent n-colorings of the 4D hypercube under the full orthogonal group of the cube (of order 2^4*4! = 384).
%C A128767 I assume this refers to colorings of the vertices of the cube. - _N. J. A. Sloane_, Apr 06 2007
%C A128767 Number of unoriented colorings of the 16 tetrahedral facets of a hyperoctahedron or of the 16 vertices of a tesseract (4-D cube) using up to n colors. Each chiral pair is counted as one when enumerating unoriented 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. - _Robert A. Russell_, Oct 03 2020
%D A128767 Banks, D. C.; Linton, S. A. & Stockmeyer, P. K. Counting Cases in Substitope Algorithms. IEEE Transactions on Visualization and Computer Graphics, Vol. 10, No. 4, pp. 371-384. 2004.
%D A128767 Perez-Aguila, Ricardo. Enumerating the Configurations in the n-Dimensional Orthogonal Polytopes Through Polya's Counting and A Concise Representation. Proceedings of the 3rd International Conference on Electrical and Electronics Engineering and XII Conference on Electrical Engineering ICEEE and CIE 2006, pp. 63-66.
%D A128767 Polya, G. & Read R. C. Combinatorial Enumeration of Groups, Graphs and Chemical Compounds. Springer-Verlag, 1987.
%H A128767 Banks, D. C.; Linton, S. A. & Stockmeyer, P. K., <a href="http://www.cs.fsu.edu/~banks/">Counting Cases in Substitope Algorithms</a>, IEEE Transactions on Visualization and Computer Graphics, Vol. 10, No. 4, pp. 371-384. 2004.
%H A128767 Perez-Aguila, Ricardo, <a href="http://ricardo.perez.aguila.googlepages.com/ricardoperez-aguila,phdthesis-orthogonalpolytopes:studyandapplication2">Orthogonal Polytopes: Study and Application</a>, PhD Thesis. Universidad de las Americas, Puebla. November, 2006.
%H A128767 Perez-Aguila, Ricardo, <a href="http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?isnumber=4017927&arnumber=4017934&count=140&index=6">Enumerating the Configurations in the n-Dimensional Orthogonal Polytopes Through Polya's Counting and A Concise Representation</a>, Proceedings of the 3rd International Conference on Electrical and Electronics Engineering and XII Conference on Electrical Engineering ICEEE and CIE 2006, pp. 63-66.
%H A128767 <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 A128767 a(n) = (1/384)*(48*n^2 + 180*n^4 + 48*n^6 + 83*n^8 + 12*n^10 + 12*n^12 + n^16)
%F A128767 G.f.: -x*(x +1)*(x^14 +384*x^13 +125020*x^12 +9439904*x^11 +213777216*x^10 +1821620108*x^9 +6527222787*x^8 +10098845160*x^7 +6527222787*x^6 +1821620108*x^5 +213777216*x^4 +9439904*x^3 +125020*x^2 +384*x +1) / (x -1)^17. [_Colin Barker_, Dec 04 2012]
%F A128767 From _Robert A. Russell_, Oct 03 2020: (Start)
%F A128767 a(n) = 1*C(n,1) + 400*C(n,2) + 130899*C(n,3) + 11230666*C(n,4) + 347919225*C(n,5) + 5158324560*C(n,6) + 43174480650*C(n,7) + 225086553300*C(n,8) + 775894225050*C(n,9) + 1831178115900*C(n,10) + 3008073915000*C(n,11) + 3439243962000*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 unoriented colorings using exactly k colors.
%F A128767 a(n) = A337952(n) - A337954(n) = (A337952(n) + A337955(n)) / 2 = A337954(n) + A337955(n).
%F A128767 (End)
%e A128767 a(2)=402 because there are 402 inequivalent 2-colorings of the 4D hypercube.
%t A128767 Table[(1/384)*( 48*n^2 + 180*n^4 + 48*n^6 + 83*n^8 + 12*n^10 + 12*n^12 + n^16),{n,30}]
%Y A128767 Cf. A337952 (oriented), A337954 (chiral), A337955 (achiral).
%Y A128767 Other elements: A331359 (tesseract edges, hyperoctahedron faces), A331355 (tesseract faces, hyperoctahedron edges), A337957 (tesseract facets, hyperoctahedron vertices).
%Y A128767 Other polychora: A000389(n+4) (4-simplex facets/vertices), A338949 (24-cell), A338965 (120-cell, 600-cell).
%Y A128767 Row 4 of A325013 (orthoplex facets, orthotope vertices).
%K A128767 nonn,easy
%O A128767 1,2
%A A128767 Ricardo Perez-Aguila (ricardo.perez.aguila(AT)gmail.com), Apr 04 2007