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A325008 Triangle read by rows: T(n,k) is the number of oriented colorings of the facets of a regular n-dimensional orthotope using exactly k colors. Row n has 2n columns. 9
1, 2, 1, 4, 9, 6, 1, 8, 30, 68, 75, 30, 1, 13, 84, 312, 735, 1020, 735, 210, 1, 19, 192, 1122, 4155, 10242, 16380, 15960, 8505, 1890, 1, 26, 381, 3322, 18285, 67679, 173936, 308056, 363825, 270900, 114345, 20790, 1, 34, 687, 8484, 66765, 352359, 1305612, 3479268, 6668865, 9035460, 8378370, 5031180, 1756755, 270270 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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. Two oriented colorings are the same if one is a rotation of the other; chiral pairs are counted as two.

Also the number of oriented 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) = A325009(n,k) + A325010(n,k) = (A325009(n,k) + A325011(n,k)) / 2 = 2*A325010(n,k) + A325011(n,k).

EXAMPLE

Triangle begins with T(1,1):

1  2

1  4   9    6

1  8  30   68    75    30

1 13  84  312   735  1020    735    210

1 19 192 1122  4155 10242  16380  15960   8505   1890

1 26 381 3322 18285 67679 173936 308056 363825 270900 114345 20790

For T(2,2)=4, there are two squares with just one edge for one color, one square with opposite edges the same color, and one square with opposite edges different colors.

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. A325009 (unoriented), A325010 (chiral), A325011 (achiral), A325004 (up to k colors).

Other n-dimensional polytopes: A325002 (simplex), A325016 (orthoplex).

Sequence in context: A077878 A128058 A163236 * A325016 A077160 A228043

Adjacent sequences:  A325005 A325006 A325007 * A325009 A325010 A325011

KEYWORD

nonn,tabf,easy

AUTHOR

Robert A. Russell, May 27 2019

STATUS

approved

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Last modified June 21 01:04 EDT 2021. Contains 345330 sequences. (Running on oeis4.)