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A340106 Triangle read by rows: T(n,k) is the number of permutations of k elements from [1..n] with longest consecutive chain size less than 3.

%I A340106 #11 Jan 20 2021 18:57:14
%S A340106 1,1,1,1,2,2,1,3,6,4,1,4,12,20,18,1,5,20,54,100,92,1,6,30,112,318,600,
%T A340106 570,1,7,42,200,768,2208,4244,4082,1,8,56,324,1570,6080,17682,34300,
%U A340106 33292,1,9,72,490,2868,13980,54552,159702,311808,304490,1,10,90,704,4830,28392,139130,545528,1604616,3147164,3086890
%N A340106 Triangle read by rows: T(n,k) is the number of permutations of k elements from [1..n] with longest consecutive chain size less than 3.
%C A340106 In a convex n-gon, the number of paths using k non-repeated vertices and fewer than 3 vertices (2 sides) in a row, except side (1,n) is unrestricted.
%F A340106 T(n,k) = A340107(n,k) + 2*O(n-1,k-1) + O(n-2,k-2), where O(n,k) = 2*(k-1)*T(n-1,k-1)/(n-1) - 2*O(n-1,k-1) + 3*O(n-2,k-2) + 2*O(n-3,k-3) + O(n-4,k-4), O(n,k)=0 for k<=1.
%e A340106 n\k   0     1      2      3      4     5     6     7     8
%e A340106 0     1
%e A340106 1     1     1
%e A340106 2     1     2      2
%e A340106 3     1     3      6      4
%e A340106 4     1     4     12     20     18
%e A340106 5     1     5     20     54    100    92
%e A340106 6     1     6     30    112    318   600    570
%e A340106 7     1     7     42    200    768  2208   4244  4082
%e A340106 8     1     8     56    324   1570  6080  17682 34300 33292
%Y A340106 Right diagonal is A095816.
%Y A340106 Cf. A338526, A338838, A338849.
%Y A340106 Cf. A340107, A340108.
%K A340106 nonn,tabl
%O A340106 0,5
%A A340106 _Xiangyu Chen_, Dec 28 2020