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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. 2
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified September 28 20:00 EDT 2021. Contains 347717 sequences. (Running on oeis4.)