

A325010


Triangle read by rows: T(n,k) is the number of chiral pairs of colorings of the facets of a regular ndimensional 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
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OFFSET

1,5


COMMENTS

Also called hypercube, ndimensional 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 SchlĂ¤fli symbol, {4,3,...,3}, of the regular ndimensional orthotope (n>1) consists of a four followed by n2 threes. Each of its 2n facets is an (n1)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 ndimensional 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..k2} binomial(jk1,j) * binomial(binomial(kj,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[jk1, j]Binomial[Binomial[kj, 2], n], {j, 0, k2}], {n, 1, 10}, {k, 1, 2n}] // Flatten


CROSSREFS

Cf. A325008 (oriented), A325009 (unoriented), A325011 (achiral), A325006 (up to k colors).
Other ndimensional 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



