login
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
4
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.
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
KEYWORD
nonn,tabl
AUTHOR
Xiangyu Chen, Nov 12 2020
STATUS
approved