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

Author

Ludovic Schwob, Apr 12 2021

Keywords

• nonn
• tabl

Comments

Polygons that differ by rotation or reflection are counted separately.

T(1,0)=0 by convention.

Examples

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

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

Crossrefs

Row sums are A307923.

Cf. A330660.

Programs

PARI

T(n)={
  local(Cache=Map());
  my(dir(p,q)=if(p=n&&qp-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<Andrew Howroyd, May 14 2021

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.

Extensions

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