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A325010 Triangle read by rows: T(n,k) is the number of chiral pairs of colorings of the facets of a regular n-dimensional orthotope using exactly k colors. Row n has 2n columns.

%I A325010 #5 May 27 2019 18:02:27
%S A325010 0,1,0,0,3,3,0,0,1,16,30,15,0,0,0,15,135,330,315,105,0,0,0,6,222,1581,
%T A325010 4410,5880,3780,945,0,0,0,1,205,3760,23604,71078,116550,107100,51975,
%U A325010 10395,0,0,0,0,120,5715,73755,427260,1351980,2552130,2962575,2079000,810810,135135
%N A325010 Triangle read by rows: T(n,k) is the number of chiral pairs of colorings of the facets of a regular n-dimensional orthotope using exactly k colors. Row n has 2n columns.
%C A325010 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. The chiral colorings of its facets come in pairs, each the reflection of the other.
%C A325010 Also the number of chiral pairs of colorings of the vertices of a regular n-dimensional orthoplex using exactly k colors.
%H A325010 Robert A. Russell, <a href="/A325010/b325010.txt">Table of n, a(n) for n = 1..132</a>
%F A325010 T(n,k) = Sum{j=0..k-2} binomial(j-k-1,j) * binomial(binomial(k-j,2),n).
%F A325010 T(n,k) = A325008(n,k) - A325009(n,k) = (A325008(n,k) - A325011(n,k)) / 2 = A325009(n,k) - A325011(n,k).
%e A325010 The triangle begins with T(1,1):
%e A325010  0 1
%e A325010  0 0 3  3
%e A325010  0 0 1 16  30   15
%e A325010  0 0 0 15 135  330   315    105
%e A325010  0 0 0  6 222 1581  4410   5880    3780     945
%e A325010  0 0 0  1 205 3760 23604  71078  116550  107100   51975   10395
%e A325010  0 0 0  0 120 5715 73755 427260 1351980 2552130 2962575 2079000 810810 135135
%e A325010 For T(2,3)=3, the three squares have the two edges with the same color adjacent.
%t A325010 Table[Sum[Binomial[j-k-1,j]Binomial[Binomial[k-j,2],n],{j,0,k-2}],{n,1,10},{k,1,2n}] // Flatten
%Y A325010 Cf. A325008 (oriented), A325009 (unoriented), A325011 (achiral), A325006 (up to k colors).
%Y A325010 Other n-dimensional polytopes: A325018 (orthoplex).
%K A325010 nonn,tabf,easy
%O A325010 1,5
%A A325010 _Robert A. Russell_, May 27 2019