A325009 Triangle read by rows: T(n,k) is the number of unoriented colorings of the facets of a regular n-dimensional orthotope using exactly k colors. Row n has 2n columns.

1, 1, 1, 4, 6, 3, 1, 8, 29, 52, 45, 15, 1, 13, 84, 297, 600, 690, 420, 105, 1, 19, 192, 1116, 3933, 8661, 11970, 10080, 4725, 945, 1, 26, 381, 3321, 18080, 63919, 150332, 236978, 247275, 163800, 62370, 10395, 1, 34, 687, 8484, 66645, 346644, 1231857, 3052008, 5316885, 6483330, 5415795, 2952180, 945945, 135135

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. Two unoriented colorings are the same if congruent; chiral pairs are counted as one.

Also the number of unoriented 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).

T(n,k) = A325009(n,k) + A325010(n,k) = 2*A325009(n,k) - A325011(n,k) = 2*A325010(n,k) + A325011(n,k).

Example

The triangle begins with T(1,1):
1  1
1  4   6    3
1  8  29   52    45    15
1 13  84  297   600   690    420    105
1 19 192 1116  3933  8661  11970  10080   4725    945
1 26 381 3321 18080 63919 150332 236978 247275 163800 62370 10395
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}], {n, 1, 10}, {k, 1, 2n}] // Flatten

Crossrefs

Cf. A325008 (oriented), A325010 (chiral), A325011 (achiral), A325005 (up to k colors).

Other n-dimensional polytopes: A007318(n,k-1) (simplex), A325017 (orthoplex).

Sequence in context: A204174 A086467 A197731 * A325017 A338147 A338143

Adjacent sequences: A325006 A325007 A325008 * A325010 A325011 A325012

Keyword

nonn,tabf,easy

Author

Robert A. Russell, May 27 2019

Status

approved