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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.

%I A343369 #20 May 26 2021 12:17:31
%S A343369 0,0,1,6,10,0,296,391,56,1,21580,28298,6132,246,0,2317884,3137098,
%T A343369 859536,70389,1012,1,349281380,490054052,158307216,19756138,711692,
%U A343369 4082,0,70651004192,102443715659,37521267472,6221752657,390266848,6782563,16368,1
%N 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.
%C A343369 Polygons that differ by rotation or reflection are counted separately.
%C A343369 T(1,0)=0 by convention.
%H A343369 Andrew Howroyd, <a href="/A343369/b343369.txt">Table of n, a(n) for n = 1..55</a> (rows 1..10)
%H A343369 Ludovic Schwob, <a href="/A343369/a343369.pdf">Illustration of T(4,k), k=0..3</a>
%H A343369 Wikipedia, <a href="https://en.wikipedia.org/wiki/Winding_number">Winding number</a>
%F A343369 T(2*n,2*n-1) = 1 and T(2*n+1,2*n) = 0 for all n>=1.
%F A343369 T(n,n-2) = 4^(n-1)-2*n for all n>=2.
%e A343369 Triangle begins:
%e A343369       0;
%e A343369       0,     1;
%e A343369       6,    10,    0;
%e A343369     296,   391,   56,   1;
%e A343369   21580, 28298, 6132, 246,   0;
%o A343369 (PARI)
%o A343369 T(n)={
%o A343369   local(Cache=Map());
%o A343369   my(dir(p,q)=if(p<n, if(q>=n&&q<p+n, 'x, 1), if(q<n&&q>p-n, 1/'x, 1)));
%o A343369   my(recurse(k,p,b) = my(hk=[k,p,b], z); if(!mapisdefined(Cache, hk, &z),
%o A343369   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)) )));
%o A343369   mapput(Cache, hk, z)); z);
%o A343369   my(p=recurse(2*n-1, 0, 0));
%o A343369   vector(n, i, polcoef(p,i-1)/if(i==1,2,1))
%o A343369 }
%o A343369 { for(n=1, 6, print(T(n))) } \\ _Andrew Howroyd_, May 14 2021
%Y A343369 Row sums are A307923.
%Y A343369 Cf. A330660.
%K A343369 nonn,tabl
%O A343369 1,4
%A A343369 _Ludovic Schwob_, Apr 12 2021
%E A343369 a(22)-a(36) from _Andrew Howroyd_, May 14 2021