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 A366541 #24 Nov 29 2024 23:50:49
%S A366541 2,4,4,5,9,6,6,14,16,8,7,20,30,25,10,8,27,50,55,36,12,9,35,77,105,91,
%T A366541 49,14,10,44,112,182,196,140,64,16,11,54,156,294,378,336,204,81,18,12,
%U A366541 65,210,450,672,714,540,285,100,20,13,77,275,660,1122,1386,1254,825,385,121,22
%N A366541 Regular triangle read by rows: T(n,k) is the number of k-facets of the bipyramid on an n-simplex base.
%C A366541 The bipyramid on an n-simplex base is the direct sum of an n-simplex and a segment. It can also be seen as two (n+1)-simplices augmented together at their base.
%C A366541 Note that all facets are simplices.
%C A366541 A deltatope is a polytope whose all cells are regular simplices (a priori not necessarily having the same size). Every polygon, being regular or not, is a 2-deltatope by definition. There are 8 convex 3-deltatopes or deltahedra (regular tetrahedron, regular octahedron, regular icosahedron, regular triangular bipyramid, regular pentagonal bipyramid and three others), 5 convex 4-deltatopes (regular 5-cell, regular 16-cell, regular 600-cell, regular tetrahedral bipyramid and regular icosahedral bipyramid) and 3 in dimension d >= 5 (regular d-simplex, regular d-orthoplex and regular bipyramid on a (d-1)-simplex base). Note that the regular orthoplex is the regular bipyramid on a hypercube base. It turns out that all cells of a deltatope are congruent (i.e., having the same size) in all nontrivial dimensions (dimension >= 3). See Dr. Richard Klitzing's answer to the Math Overflow question "4-polytopes with only one kind of regular facet" for dimension 4, and Gjergji Zaimi's answer to the question "Convex deltahedra in higher dimensions" for dimension >= 5.
%C A366541 More generally, a convex polytope whose all cells are regular polytopes of the same kind is either regular or a deltatope. See the article of Roswitha Blind.
%C A366541 The symmetry group of the bipyramid on an n-simplex base, generated by the symmetries of the n-simplex and the vertical reflexion that commute, is S_{n+1} X C_2 (with Coxeter notation [2,3^(n-1)]). See the Math Stack Exchange link.
%H A366541 Jianing Song, <a href="/A366541/b366541.txt">Table of n, a(n) for n = 0..5150</a> (Rows n = 0..100)
%H A366541 Math Overflow, <a href="https://mathoverflow.net/q/342224">4-polytopes with only one kind of regular facet</a>.
%H A366541 Math Overflow, <a href="https://mathoverflow.net/q/149185">Convex deltahedra in higher dimensions</a>.
%H A366541 Mathematics Stack Exchange, <a href="https://math.stackexchange.com/q/4059195">Group symmetries of a trigonal bipyramidal molecule</a>.
%H A366541 Roswitha Blind, Jürgen Tölke and Jörg M. Wills, <a href="https://doi.org/10.1007/978-3-0348-5765-9_10">Konvexe Polytope mit regulären Facetten im R^n (n>=4)</a> (in German), Contributions to Geometry: Proceedings of the Geometry-Symposium held in Siegen June 28, 1978 to July 1, 1978.
%F A366541 T(n,k) = 2*(number of k-facets of the (n+1)-simplex) - (number of k-facets of the n-simplex) = 2*binomial(n+2,k+1) - binomial(n+1,k+1) for 0 <= k <= n-1; T(n,n) = 2*(number of n-facets of the (n+1)-simplex) - (2 cells as base) = 2*binomial(n+2,n+1) - 2 = 2*(n+1).
%e A366541 The triangle T(n, k) begins:
%e A366541 n\k 0 1 2 3 4 5 6 7 8 9 10
%e A366541 0 2
%e A366541 1 4 4
%e A366541 2 5 9 6
%e A366541 3 6 14 16 8
%e A366541 4 7 20 30 25 10
%e A366541 5 8 27 50 55 36 12
%e A366541 6 9 35 77 105 91 49 14
%e A366541 7 10 44 112 182 196 140 64 16
%e A366541 8 11 54 156 294 378 336 204 81 18
%e A366541 9 12 65 210 450 672 714 540 285 100 20
%e A366541 10 13 77 275 660 1122 1386 1254 825 385 121 22
%e A366541 n = 0: the segment has 2 vertices;
%e A366541 n = 1: the quadrilateral has 4 vertices and 4 sides;
%e A366541 n = 2: the triangular bipyramid has 5 vertices, 9 edges and 6 faces;
%e A366541 n = 3: the tetrahedral bipyramid has 6 vertices, 14 edges, 16 faces and 8 cells.
%o A366541 (PARI) T(n,k) = if(k<n, 2*binomial(n+2,k+1) - binomial(n+1,k+1), 2*(n+1))
%Y A366541 A014410(n+1,k) is the number of k-facets of the n-simplex.
%K A366541 nonn,tabl,easy
%O A366541 0,1
%A A366541 _Jianing Song_, Oct 12 2023