A045629 - OEIS

%I #24 Oct 08 2017 05:18:54

%S 1,1,2,3,7,15,44,128,415,1367,4654,16080,56450,200170,716728,2585850,

%T 9393119,34319667,126047906,465076160,1723097066,6407856892,

%U 23910271224,89493903438,335912741682,1264106399934,4768448177636,18027218147818

%N Number of 2n-bead black-white complementable necklaces with n black beads.

%H Andrew Howroyd, <a href="/A045629/b045629.txt">Table of n, a(n) for n = 0..200</a>

%H <a href="/index/Ne#necklaces">Index entries for sequences related to necklaces</a>

%F a(n) = (1/2n) * Sum_{d|n} (phi(n/d)*C(2d-1, d-1) + phi(2n/d)*2^(d-1)). - _Christian G. Bower_

%F a(n) ~ 4^(n-1) / (sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Oct 08 2017

%t a[n_] := If[n == 0, 1, (1/(2*n))*DivisorSum[n, EulerPhi[n/#1]*Binomial[2*#1 - 1, #1 - 1] + EulerPhi[2*(n/#1)]*2^(#1 - 1)&]];

%t Table[a[n], {n, 0, 27}] (* _Jean-François Alcover_, Oct 08 2017, translated from PARI *)

%o (PARI) a(n) = if(n==0, 1, (1/(2*n)) * sumdiv(n, d, eulerphi(n/d)*binomial(2*d-1, d-1) + eulerphi(2*n/d)*2^(d-1))); \\ _Andrew Howroyd_, Sep 27 2017

%Y Cf. A006840.

%K nonn

%O 0,3

%A _David W. Wilson_