A330662 Triangle read by rows: T(n,k) is the number of polygons with 2*n sides, of which k run through the center of a circle, on the circumference of which the 2*n vertices of the polygon are arranged at equal spacing.

0, 0, 1, 1, 0, 2, 16, 24, 12, 8, 744, 960, 576, 192, 48, 56256, 69120, 39360, 13440, 2880, 384, 6385920, 7580160, 4204800, 1420800, 316800, 46080, 3840, 1018114560, 1178956800, 642539520, 216115200, 49190400, 7741440, 806400, 46080

Offset

0,6

Comments

Rotations and reflections are counted separately.

By "2*n-sided polygons" we mean the polygons that can be drawn by connecting 2*n equally spaced points on a circle.

T(0,0)=0 and T(0,1)=1 by convention.

The sequence is limited to even-sided polygons, since all odd-sided polygons have no side passing through the center.

Links

Ludovic Schwob, Table of n, a(n) for n = 0..494

Ludovic Schwob, Illustration of T(3,k), 0≤k≤3.

Formula

T(n,n) = 2^(n-1) * (n-1)! for all n >= 1.

T(n,0) = A307923(n) for all n>=1.

T(n,k) = binomial(n,k)* Sum_{i=k..n} (-1)^(i-k)*binomial(n-k,i-k)*(2n-1-i)!*2^(i-1), for n>=2 and 0<=k<=n.

Example

Triangle begins:
    0;
    0,   1;
    1,   0,   2;
   16,  24,  12,   8;
  744, 960, 576, 192, 48;

Maple

T := (n, k) -> `if`(n<2, k, 2^(k-1)*binomial(n, k)*(2*n-k-1)!*hypergeom([k-n], [k-2*n+ 1], -2)):
seq(seq(simplify(T(n, k)), k=0..n), n=0..7); # Peter Luschny, Jan 07 2020

Crossrefs

Row sums give A001710(2*n-1) (number of polygons with 2*n sides).

Cf. A000165 (diagonal).

Star polygons: A014106, A055684, A102302.

Cf. A309318.

Sequence in context: A333475 A330611 A228158 * A337426 A295822 A212765

Adjacent sequences: A330659 A330660 A330661 * A330663 A330664 A330665

Keyword

nonn,tabl

Author

Ludovic Schwob, Dec 23 2019

Status

approved