This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%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