A343369 Triangle read by rows: T(n,k) is the number of polygons formed by connecting the vertices of a regular 2n-gon such that the winding number around the center is k and with no side passing through the center.

0, 0, 1, 6, 10, 0, 296, 391, 56, 1, 21580, 28298, 6132, 246, 0, 2317884, 3137098, 859536, 70389, 1012, 1, 349281380, 490054052, 158307216, 19756138, 711692, 4082, 0, 70651004192, 102443715659, 37521267472, 6221752657, 390266848, 6782563, 16368, 1

Offset

1,4

Comments

Polygons that differ by rotation or reflection are counted separately.

T(1,0)=0 by convention.

Links

Andrew Howroyd, Table of n, a(n) for n = 1..55 (rows 1..10)

Ludovic Schwob, Illustration of T(4,k), k=0..3

Wikipedia, Winding number

Formula

T(2*n,2*n-1) = 1 and T(2*n+1,2*n) = 0 for all n>=1.

T(n,n-2) = 4^(n-1)-2*n for all n>=2.

Example

Triangle begins:
      0;
      0,     1;
      6,    10,    0;
    296,   391,   56,   1;
  21580, 28298, 6132, 246,   0;

Prog

(PARI)
T(n)={
  local(Cache=Map());
  my(dir(p, q)=if(p<n, if(q>=n&&q<p+n, 'x, 1), if(q<n&&q>p-n, 1/'x, 1)));
  my(recurse(k, p, b) = my(hk=[k, p, b], z); if(!mapisdefined(Cache, hk, &z),
  z = if(k==0, p<>n, sum(q=1, 2*n-1, if(!bittest(b, q) && (q-p)%n, dir(p, q)*self()(k-1, q, b+(1<<q)) )));
  mapput(Cache, hk, z)); z);
  my(p=recurse(2*n-1, 0, 0));
  vector(n, i, polcoef(p, i-1)/if(i==1, 2, 1))
}
{ for(n=1, 6, print(T(n))) } \\ Andrew Howroyd, May 14 2021

Crossrefs

Row sums are A307923.

Cf. A330660.

Sequence in context: A202951 A316633 A295052 * A052193 A347632 A144763

Adjacent sequences: A343366 A343367 A343368 * A343370 A343371 A343372

Keyword

nonn,tabl

Author

Ludovic Schwob, Apr 12 2021

Extensions

a(22)-a(36) from Andrew Howroyd, May 14 2021

Status

approved