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%I #19 Sep 11 2022 09:29:39
%S 1,1,1,1,2,2,1,3,6,0,1,4,12,0,16,1,5,20,30,80,60,1,6,30,84,264,480,
%T 456,1,7,42,168,672,1890,3612,3458,1,8,56,288,1424,5440,15744,30352,
%U 29296,1,9,72,450,2664,12870,50004,145656,283104,275166,1,10,90,660,4560,26640,130080,508060,1488960,2909700,2843980
%N Triangle read by rows: T(n,k) is the number of permutations of k elements from [1..n] in a circle with longest consecutive chain size less than 3, when 1 and n are considered to be consecutive, and rotations are considered to be distinct.
%C In a convex n-gon, the number of cycles using k non-repeated vertices and fewer than 3 vertices (2 sides) in a row.
%F T(n,k) = n*(5*A340106(n-1,k-1) - 2*Z(n,k) - Z(n-1,k-1) - 2*S(n,k) - 2*S(n-2,k-2)) except for T(n,0)=1, where S(n,k) = 2*A340106(n-1,k-1) - 2*A340106(n-2,k-2) + S(n-3,k-3), S(n,k)=0 for k <= 0, Z(n,k) = 2*A340106(n-1,k-1) - S(n,k) - V(n-1,k-1), Z(n,k)=0 for k <= 0, V(n,k) = Z(n-1,k-1) - V(n-1,k-1), V(n,k)=0 for k <= 0 except for V(2,2)=2.
%e n\k 0 1 2 3 4 5 6 7 8
%e 0 1
%e 1 1 1
%e 2 1 2 2
%e 3 1 3 6 0
%e 4 1 4 12 0 16
%e 5 1 5 20 30 80 60
%e 6 1 6 30 84 264 480 456
%e 7 1 7 42 168 672 1890 3612 3458
%e 8 1 8 56 288 1424 5440 15744 30352 29296
%Y Cf. A338526, A338838, A338849.
%Y Cf. A340106, A340107.
%K nonn,tabl
%O 0,5
%A _Xiangyu Chen_, Dec 28 2020
%E Terms of column 2 corrected by _Xiangyu Chen_, Aug 19 2022
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