A330618 Triangle read by rows: T(n,k) is the number of n-bead necklaces using exactly k colors with no adjacent beads having the same color.
0, 0, 1, 0, 0, 2, 0, 1, 3, 6, 0, 0, 6, 24, 24, 0, 1, 11, 80, 180, 120, 0, 0, 18, 240, 960, 1440, 720, 0, 1, 33, 696, 4410, 11340, 12600, 5040, 0, 0, 58, 1960, 18760, 73920, 137760, 120960, 40320, 0, 1, 105, 5508, 76368, 433944, 1209600, 1753920, 1270080, 362880
Offset: 1
Examples
Triangle begins: 0; 0, 1; 0, 0, 2; 0, 1, 3, 6; 0, 0, 6, 24, 24; 0, 1, 11, 80, 180, 120; 0, 0, 18, 240, 960, 1440, 720; 0, 1, 33, 696, 4410, 11340, 12600, 5040; 0, 0, 58, 1960, 18760, 73920, 137760, 120960, 40320; ...
Links
- Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)
Programs
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PARI
\\ here U(n,k) is A208535(n,k) for n > 1. U(n, k)={sumdiv(n, d, eulerphi(n/d)*(k-1)^d)/n - if(n%2, k-1)} T(n,k)={sum(j=1, k, (-1)^(k-j)*binomial(k,j)*U(n,j))}
Formula
T(n,k) = Sum_{j=1..k} (-1)^(k-j)*binomial(k,j)*A208535(n,j) for n > 1.
T(n,n) = (n-1)! for n > 1.
Comments