A325001 Array read by descending antidiagonals: A(n,k) is the number of achiral colorings of the facets (or vertices) of a regular n-dimensional simplex using up to k colors.

1, 2, 1, 3, 4, 1, 4, 9, 5, 1, 5, 16, 15, 6, 1, 6, 25, 34, 21, 7, 1, 7, 36, 65, 56, 28, 8, 1, 8, 49, 111, 125, 84, 36, 9, 1, 9, 64, 175, 246, 210, 120, 45, 10, 1, 10, 81, 260, 441, 461, 330, 165, 55, 11, 1, 11, 100, 369, 736, 917, 792, 495, 220, 66, 12, 1

Offset: 1

Author

Robert A. Russell, Mar 23 2019

Keywords

• nonn
• tabl

Comments

For n=1, the figure is a line segment with two vertices. For n=2, the figure is a triangle with three edges. For n=3, the figure is a tetrahedron with four triangular faces. The Schläfli symbol, {3,...,3}, of the regular n-dimensional simplex consists of n-1 threes. Each of its n+1 facets is a regular (n-1)-dimensional simplex. An achiral coloring is the same as its reflection.

Examples

			The array begins with A(1,1):
  1  2  3   4   5    6    7     8     9    10    11     12     13 ...
  1  4  9  16  25   36   49    64    81   100   121    144    169 ...
  1  5 15  34  65  111  175   260   369   505   671    870   1105 ...
  1  6 21  56 125  246  441   736  1161  1750  2541   3576   4901 ...
  1  7 28  84 210  461  917  1688  2919  4795  7546  11452  16848 ...
  1  8 36 120 330  792 1715  3424  6399 11320 19118  31032  48672 ...
  1  9 45 165 495 1287 3003  6434 12861 24265 43593  75087 124683 ...
  1 10 55 220 715 2002 5005 11440 24309 48610 92323 167740 293215 ...
  ...
For A(2,2)=4, the triangle may have 0, 1, 2, or 3 edges of one color.
		

Links

• Robert A. Russell, Table of n, a(n) for n = 1..1275

Crossrefs

Cf. A324999 (oriented), A325000 (unoriented), A325000(n,k-n) (chiral), A325003 (exactly k colors), A327086 (edges, ridges), A337886 (faces, peaks), A325007 (orthotope facets, orthoplex vertices), A325015 (orthoplex facets, orthotope vertices).

Rows 1-4 are A000027, A000290, A006003, A132366(n-1).

Column 2 is A162880.

Programs

Mathematica

Table[Binomial[d+1,n+1] - Binomial[d+1-n,n+1], {d,1,15}, {n,1,d}] // Flatten

Formula

A(n,k) = binomial(n+k,n+1) - binomial(k,n+1).

A(n,k) = Sum_{j=1..n} A325003(n,j) * binomial(k,j).

A(n,k) = 2*A325000(n,k) - A324999(n,k) = A324999(n,k) - 2*A325000(n,k-n) = A325000(n,k) - A325000(n,k-n).

G.f. for row n: (x - x^(n+1)) / (1-x)^(n+2).

Linear recurrence for row n: A(n,k) = Sum_{j=1..n+1} -binomial(j-n-2,j) * A(n,k-j).

G.f. for column k: (1 - (1-x^2)^k) / (x*(1-x)^k).