%I #34 Oct 15 2017 12:48:15
%S 2,4,4,6,12,8,8,24,32,16,10,40,80,80,32,12,60,160,240,192,64,14,84,
%T 280,560,672,448,128,16,112,448,1120,1792,1792,1024,256,18,144,672,
%U 2016,4032,5376,4608,2304,512,20,180,960,3360,8064,13440,15360,11520,5120
%N Triangle read by rows: T(n,k) = number of k-dimensional elements in an n-dimensional cross-polytope, n>=1, 0<=k<n.
%C It appears that this is 2*A193862 (but with a different offset) and that the sum of terms of the n-th row is A024023(n) = 3^n - 1. - _Michel Marcus_, Sep 29 2016
%D H. S. M. Coxeter, Regular Polytopes, Third Edition, Dover Publications, 1973, ISBN 9780486141589.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Cross-polytope">Cross-polytope</a>.
%H D. A. Zaitsev, <a href="https://doi.org/10.1016/j.tcs.2016.11.002">A generalized neighborhood for cellular automata</a>, Theoretical Computer Science, 666 (2017), 21-35.
%F a(n, k) = 2^(k+1) * binomial(n, k+1) (cf. Coxeter, 1973, formula 7.22).
%F a(n,k) = A182059(n,k) = A013609(n,k) . - _R. J. Mathar_, May 03 2017
%e T(4, 1..4) = 8, 24, 32, 16, because the 16-cell has 8 0-faces (vertices), 24 1-faces (edges), 32 2-faces (faces) and 16 3-faces (cells).
%e Triangle starts
%e 2
%e 4, 4
%e 6, 12, 8
%e 8, 24, 32, 16
%e 10, 40, 80, 80, 32
%e 12, 60, 160, 240, 192, 64
%e 14, 84, 280, 560, 672, 448, 128
%e 16, 112, 448, 1120, 1792, 1792, 1024, 256
%e 18, 144, 672, 2016, 4032, 5376, 4608, 2304, 512
%e 20, 180, 960, 3360, 8064, 13440, 15360, 11520, 5120, 1024
%t Table[2^(k + 1) Binomial[n, k + 1], {n, 10}, {k, 0, n - 1}] // Flatten (* _Michael De Vlieger_, Sep 25 2016 *)
%o (PARI) T(n, k) = 2^(k+1)*binomial(n, k+1)
%o trianglerows(n) = for(x=1, n, for(y=0, x-1, print1(T(x, y), ", ")); print(""))
%o trianglerows(10) \\ print initial 10 rows of triangle
%Y Cf. A038207 (hypercube), A135278 (simplex).
%Y Rows: A005843(n), A046092(n), A130809(n+2), A130810(n+3).
%Y Columns: A000079(n), A001787(n), A001788(n), A001789(n+3).
%K nonn,tabl,easy
%O 1,1
%A _Felix Fröhlich_, Sep 24 2016