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A342968 Irregular triangle read by rows: T(n,k) is the number of n+2-sided polygons with the property that one makes k turns on itself while following its edges.

%I A342968 #25 Jun 13 2021 07:19:00
%S A342968 1,0,1,2,1,5,6,1,24,28,8,119,183,57,1,832,1209,432,47,6255,9514,3760,
%T A342968 630,1,54380,82636,36352,7828,244,515284,812714,383648,94997,7756,1,
%U A342968 5454624,8727684,4377888,1243482,153536,1186
%N A342968 Irregular triangle read by rows: T(n,k) is the number of n+2-sided polygons with the property that one makes k turns on itself while following its edges.
%C A342968 Polygons that differ by rotation or reflection are counted separately.
%C A342968 By "n+2-sided polygons" we mean the polygons that can be drawn by connecting n+2 equally spaced points on a circle (possibly self-intersecting).
%C A342968 T(0,0)=1 by convention.
%C A342968 To compute the number of turns, follow the edges of the polygon, and add the angles of rotation: positive if turning left, negative if turning right. Then take the absolute value of the sum (see illustration).
%H A342968 Ludovic Schwob, <a href="/A342968/a342968.pdf">Illustration of T(5,k), 0 <= k <= 3</a>
%F A342968 T(2*n-1,n)=1 for all n >= 1: the only solution is the polygon with Schläfli symbol {2*n+1/n}.
%e A342968 Triangle begins:
%e A342968      1;
%e A342968      0,    1;
%e A342968      2,    1;
%e A342968      5,    6,    1;
%e A342968     24,   28,    8;
%e A342968    119,  183,   57,   1;
%Y A342968 Row sums give A001710(n+1) (number of polygons with n+2 sides).
%K A342968 nonn,tabf,more
%O A342968 0,4
%A A342968 _Ludovic Schwob_, Apr 01 2021