A338849 Triangle read by rows: T(n,k) is the number of permutations of k elements from [1..n] in a circle where adjacent values cannot be consecutive modulo n, rotations are distinct

1, 1, 1, 1, 2, 0, 1, 3, 0, 0, 1, 4, 4, 0, 0, 1, 5, 10, 0, 0, 10, 1, 6, 18, 12, 24, 60, 36, 1, 7, 28, 42, 112, 280, 420, 322, 1, 8, 40, 96, 336, 1040, 2400, 3696, 2832, 1, 9, 54, 180, 792, 3060, 9540, 22428, 35280, 27954, 1, 10, 70, 300, 1600, 7540, 29880, 95340, 229280, 369540, 299260

Offset

0,5

Comments

In a convex n-gon, the number of paths using k-1 diagonals and k non-repeated vertices, start and end vertices are not connected by a side.

Links

Table of n, a(n) for n=0..65.

Formula

T(n,k) = n*(A338526(n-1,k-1)-2*S(n-1,k-1)+Z2(n-1,k-1)) for k>0 except T(2,2)=0, T(n,0)=1, where Z2(n,k) = Z1(n,k) except Z2(n,1)=2, where Z1(n,k) = S(n-1,k-1)-Z(n-1,k-1) for k>0 except Z1(2,2)=0, Z1(n,0)=0, where S(n,k) = 2*A338526(n-1,k-1)-S(n-1,k-1) for k>0, S(n,0)=0.

Example

n\k  0    1    2    3    4    5    6    7    8
0    1
1    1    1
2    1    2    0
3    1    3    0    0
4    1    4    4    0    0
5    1    5    10   0    0    10
6    1    6    18   12   24   60   36
7    1    7    28   42   112  280  420  322
8    1    8    40   96   336  1040 2400 3696 2832

Crossrefs

Cf. A011973, A034807, A338526, A338838, A000166.

Sequence in context: A004199 A062283 A136493 * A338838 A132213 A202502

Adjacent sequences: A338846 A338847 A338848 * A338850 A338851 A338852

Keyword

nonn,tabl

Author

Xiangyu Chen, Nov 12 2020

Status

approved