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%I #44 Sep 30 2020 05:01:20
%S 1,4,91,5106,510312,79419180,17758541160,5397245416080,
%T 2140495978440960,1073686615987184640,664582969579048732800,
%U 497566995304189676342400,443212653988584642449548800,463237380681508395323231270400,561422444732790213860755013145600,780983354978825959061219179885824000
%N a(n) is the number of 2n-bead necklaces with exactly n different colored beads.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Necklace_(combinatorics)">Necklace combinatorics</a>
%F 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.
%F 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.
%e a(2) = 4, corresponding to the necklaces WBBB, WBWB, WWBB, and WWWB.
%t Table[n! * (StirlingS2[2*n, n] + 1) / (2*n), {n, 1, 16}] (* _Amiram Eldar_, Sep 25 2020 *)
%o (PARI) T(n,k) = (k!/n) * sumdiv(n,d, eulerphi(d) * stirling(n/d, k,2)); \\ A087854
%o vector(22,n,T(2*n,n)) \\ _Joerg Arndt_, Sep 25 2020
%Y Cf. A087854, A007820.
%K nonn
%O 1,2
%A _Yves-Loic Martin_, Sep 24 2020
%E Terms a(6) and beyond from _Joerg Arndt_, Sep 25 2020
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