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 A325011 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. 9
 1, 0, 1, 4, 3, 0, 1, 8, 28, 36, 15, 0, 1, 13, 84, 282, 465, 360, 105, 0, 1, 19, 192, 1110, 3711, 7080, 7560, 4200, 945, 0, 1, 26, 381, 3320, 17875, 60159, 126728, 165900, 130725, 56700, 10395, 0, 1, 34, 687, 8484, 66525, 340929, 1158102, 2624748, 3964905, 3931200, 2453220, 873180, 135135, 0 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS 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. Also the number of achiral colorings of the vertices of a regular n-dimensional orthoplex using exactly k colors. LINKS Robert A. Russell, Table of n, a(n) for n = 1..132 FORMULA 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). T(n,k) = 2*A325009(n,k) - A325008(n,k) = A325008(n,k) - 2*A325010(n,k) = A325009(n,k) - A325010(n,k). EXAMPLE Table begins with T(1,1): 1  0 1  4   3    0 1  8  28   36    15     0 1 13  84  282   465   360    105      0 1 19 192 1110  3711  7080   7560   4200    945     0 1 26 381 3320 17875 60159 126728 165900 130725 56700 10395 0 For T(2,3)=3, each of the three chiral pairs has two opposite edges with the same color. MATHEMATICA 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 CROSSREFS Cf. A325008 (oriented), A325009 (unoriented), A325010 (chiral), A325007 (up to k colors). Other n-dimensional polytopes: A325003 (simplex), A325019 (orthoplex). Sequence in context: A131106 A298739 A346366 * A294188 A331956 A325019 Adjacent sequences:  A325008 A325009 A325010 * A325012 A325013 A325014 KEYWORD nonn,tabf,easy AUTHOR Robert A. Russell, May 27 2019 STATUS approved

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Last modified September 24 19:27 EDT 2021. Contains 347651 sequences. (Running on oeis4.)