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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. 8
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, 4410, 5880, 3780, 945, 0, 0, 0, 1, 205, 3760, 23604, 71078, 116550, 107100, 51975, 10395, 0, 0, 0, 0, 120, 5715, 73755, 427260, 1351980, 2552130, 2962575, 2079000, 810810, 135135 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

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. The chiral colorings of its facets come in pairs, each the reflection of the other.

Also the number of chiral pairs of 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-2} binomial(j-k-1,j) * binomial(binomial(k-j,2),n).

T(n,k) = A325008(n,k) - A325009(n,k) = (A325008(n,k) - A325011(n,k)) / 2 = A325009(n,k) - A325011(n,k).

EXAMPLE

The triangle begins with T(1,1):

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  4410   5880    3780     945

0 0 0  1 205 3760 23604  71078  116550  107100   51975   10395

0 0 0  0 120 5715 73755 427260 1351980 2552130 2962575 2079000 810810 135135

For T(2,3)=3, the three squares have the two edges with the same color adjacent.

MATHEMATICA

Table[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), A325011 (achiral), A325006 (up to k colors).

Other n-dimensional polytopes: A325018 (orthoplex).

Sequence in context: A000876 A287712 A287784 * A297876 A298139 A298087

Adjacent sequences:  A325007 A325008 A325009 * A325011 A325012 A325013

KEYWORD

nonn,tabf,easy

AUTHOR

Robert A. Russell, May 27 2019

STATUS

approved

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Last modified July 30 03:08 EDT 2021. Contains 346347 sequences. (Running on oeis4.)