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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.

%I A330662 #47 Jun 13 2021 03:30:27
%S A330662 0,0,1,1,0,2,16,24,12,8,744,960,576,192,48,56256,69120,39360,13440,
%T A330662 2880,384,6385920,7580160,4204800,1420800,316800,46080,3840,
%U A330662 1018114560,1178956800,642539520,216115200,49190400,7741440,806400,46080
%N 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.
%C A330662 Rotations and reflections are counted separately.
%C A330662 By "2*n-sided polygons" we mean the polygons that can be drawn by connecting 2*n equally spaced points on a circle.
%C A330662 T(0,0)=0 and T(0,1)=1 by convention.
%C A330662 The sequence is limited to even-sided polygons, since all odd-sided polygons have no side passing through the center.
%H A330662 Ludovic Schwob, <a href="/A330662/b330662.txt">Table of n, a(n) for n = 0..494</a>
%H A330662 Ludovic Schwob, <a href="/A330662/a330662.pdf">Illustration of T(3,k), 0≤k≤3</a>.
%F A330662 T(n,n) = 2^(n-1) * (n-1)! for all n >= 1.
%F A330662 T(n,0) = A307923(n) for all n>=1.
%F A330662 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.
%e A330662 Triangle begins:
%e A330662     0;
%e A330662     0,   1;
%e A330662     1,   0,   2;
%e A330662    16,  24,  12,   8;
%e A330662   744, 960, 576, 192, 48;
%p A330662 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)):
%p A330662 seq(seq(simplify(T(n,k)), k=0..n),n=0..7); # _Peter Luschny_, Jan 07 2020
%Y A330662 Row sums give A001710(2*n-1) (number of polygons with 2*n sides).
%Y A330662 Cf. A000165 (diagonal).
%Y A330662 Star polygons: A014106, A055684, A102302.
%Y A330662 Cf. A309318.
%K A330662 nonn,tabl
%O A330662 0,6
%A A330662 _Ludovic Schwob_, Dec 23 2019