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%I #5 May 27 2019 18:02:54
%S 1,0,1,4,3,0,1,8,28,36,15,0,1,13,84,282,465,360,105,0,1,19,192,1110,
%T 3711,7080,7560,4200,945,0,1,26,381,3320,17875,60159,126728,165900,
%U 130725,56700,10395,0,1,34,687,8484,66525,340929,1158102,2624748,3964905,3931200,2453220,873180,135135,0
%N Triangle read by rows: T(n,k) is the number of achiral colorings of the facets of a regular n-dimensional orthotope using exactly k colors. Row n has 2n columns.
%C Also called hypercube, n-dimensional cube, and measure polytope. For n=1, the figure is a line segment with two vertices. For n=2 the figure is a square with four edges. For n=3 the figure is a cube with six square faces. For n=4, the figure is a tesseract with eight cubic facets. The Schläfli symbol, {4,3,...,3}, of the regular n-dimensional orthotope (n>1) consists of a four followed by n-2 threes. Each of its 2n facets is an (n-1)-dimensional orthotope. An achiral coloring is identical to its reflection.
%C Also the number of achiral colorings of the vertices of a regular n-dimensional orthoplex using exactly k colors.
%H Robert A. Russell, <a href="/A325011/b325011.txt">Table of n, a(n) for n = 1..132</a>
%F T(n,k) = Sum_{j=0..k-1} binomial(-j-2,k-j-1) * binomial(n + binomial(j+2,2)-1, n) - Sum_{j=0..k-2} binomial(j-k-1,j) * binomial(binomial(k-j,2),n).
%F T(n,k) = 2*A325009(n,k) - A325008(n,k) = A325008(n,k) - 2*A325010(n,k) = A325009(n,k) - A325010(n,k).
%e Table begins with T(1,1):
%e 1 0
%e 1 4 3 0
%e 1 8 28 36 15 0
%e 1 13 84 282 465 360 105 0
%e 1 19 192 1110 3711 7080 7560 4200 945 0
%e 1 26 381 3320 17875 60159 126728 165900 130725 56700 10395 0
%e For T(2,3)=3, each of the three chiral pairs has two opposite edges with the same color.
%t Table[Sum[Binomial[-j-2,k-j-1] Binomial[n + Binomial[j+2,2]-1, n], {j,0,k-1}] - Sum[Binomial[j-k-1,j] Binomial[Binomial[k-j,2],n],{j,0,k-2}], {n,1,10},{k,1,2n}] // Flatten
%Y Cf. A325008 (oriented), A325009 (unoriented), A325010 (chiral), A325007 (up to k colors).
%Y Other n-dimensional polytopes: A325003 (simplex), A325019 (orthoplex).
%K nonn,tabf,easy
%O 1,4
%A _Robert A. Russell_, May 27 2019
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