A337827 a(n) is the number of 2n-bead necklaces with exactly n different colored beads.
1, 4, 91, 5106, 510312, 79419180, 17758541160, 5397245416080, 2140495978440960, 1073686615987184640, 664582969579048732800, 497566995304189676342400, 443212653988584642449548800, 463237380681508395323231270400, 561422444732790213860755013145600, 780983354978825959061219179885824000
Offset: 1
Keywords
Examples
a(2) = 4, corresponding to the necklaces WBBB, WBWB, WWBB, and WWWB.
Links
- Wikipedia, Necklace combinatorics
Programs
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Mathematica
Table[n! * (StirlingS2[2*n, n] + 1) / (2*n), {n, 1, 16}] (* Amiram Eldar, Sep 25 2020 *) -
PARI
T(n,k) = (k!/n) * sumdiv(n,d, eulerphi(d) * stirling(n/d, k,2)); \\ A087854 vector(22,n,T(2*n,n)) \\ Joerg Arndt, Sep 25 2020
Formula
a(n) = A087854(2*n,n) = (n!/(2*n)) * Sum_{d|2*n} phi(d) * S2(2*n/d, n) where S2(n,k) are the Stirling numbers of the second kind.
a(n) = (n!/(2*n))*(S2(2*n, n)+1) since S2(n, n) = 1 and S2(2*n/d, n) = 0 if d>2.
Extensions
Terms a(6) and beyond from Joerg Arndt, Sep 25 2020