Necklaces vs Bracelets

[Antti Karttunen]

Frank Ruskey wrote:
> 
> Burnside's Lemma gives:
> 
> Length n binary bracelets with k black beads = (T(n,k)+s)/2 where
> 
> s = C( n/2,     r/2     )  if n = 0, r = 0 mod 2
>     C( (n-2)/2, (r-1)/2 )  if n = 0, r = 1 mod 2
>     C( (n-1)/2, r/2     )  if n = 1, r = 0 mod 2
>     C( (n-1)/2, (r-2)/2 )  if n = 1, r = 1 mod 2
> 
> The expression for s can undoubtedly be simplified with some
> floors, but the above expression is closer to the derivation.

I reckon that Frank meant above with "r" the same as with "k",
and T(n,k) is A047996(n,k).

Anyways, combined with the proof based on Raney's lemma
that A047966(2n+1,n) = A000108(n) this is enough to prove
that A007123(n) = (A000108(n)+A001405(n))/2.

(And you can forget my previous long-winding mail...)

-- Antti