A330341 Triangle read by rows: T(n,k) is the number of n-bead bracelets using exactly k colors with no adjacent beads having the same color.
0, 0, 1, 0, 0, 1, 0, 1, 3, 3, 0, 0, 3, 12, 12, 0, 1, 10, 46, 90, 60, 0, 0, 9, 120, 480, 720, 360, 0, 1, 27, 384, 2235, 5670, 6300, 2520, 0, 0, 29, 980, 9380, 36960, 68880, 60480, 20160, 0, 1, 75, 2904, 38484, 217152, 604800, 876960, 635040, 181440
Offset: 1
Examples
Triangle begins: 0; 0, 1; 0, 0, 1; 0, 1, 3, 3; 0, 0, 3, 12, 12; 0, 1, 10, 46, 90, 60; 0, 0, 9, 120, 480, 720, 360; 0, 1, 27, 384, 2235, 5670, 6300, 2520; 0, 0, 29, 980, 9380, 36960, 68880, 60480, 20160; ...
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 A208544(n, k) for n > 1. U(n, k) = (sumdiv(n, d, eulerphi(n/d)*(k-1)^d)/n + if(n%2, 1-k, k*(k-1)^(n/2)/2))/2; 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)*A208544(n,j) for n > 1.
Comments