A082936 a(n) = (1/(3*n))*Sum_{d|n, d even} phi(2*n/d)*binomial(3d/2,d).

1, 1, 3, 10, 43, 201, 1038, 5538, 30667, 173593, 1001603, 5864750, 34769374, 208267320, 1258579654, 7663720710, 46976034379, 289628805623, 1794932468571, 11175157356522, 69864075597643, 438403736549145, 2760351032959050, 17433869214973754, 110420300879752990

Offset: 0

Author

N. J. A. Sloane, May 26 2003

Keywords

• nonn

Comments

a(n) = number of necklaces of n white beads and 2n black beads. - David Callan, Mar 28 2004

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• M. Bousquet and C. Lamathe, Enumeration of solid trees according to edge number and edge degree distribution, Discr. Math., 298 (2005), 115-141.
• M. Isachenkov, I. Kirsch, V. Schomerus, Chiral Primaries in Strange Metals, arXiv preprint arXiv:1403.6857 [hep-th], 2014.

Crossrefs

Cf. A003239.

Column k=2 of A261494.

Programs

Maple

with(numtheory): f := proc(n) local t1,d; t1 := 0; for d from 1 to n do if n mod d = 0 then if d mod 2 = 0 then t1 := t1+phi(n/d)*binomial(3*d/2,d) fi; fi; od; 2*t1/(3*n); end; # use with n even

Mathematica

a[n_] := DivisorSum[n, EulerPhi[n/#]*Binomial[3#, #]&]/(3n); a[0] = 1; Array[a, 30, 0] (* Jean-François Alcover, Dec 02 2015 *)

PARI

C(n, k)=binomial(n,k);
a(n) = if(n<=0, n==0, sumdiv(n, d, eulerphi(n/d) * C(3*d,d)) / (3*n) );
/* or, second formula: */
/* a(n) = if(n<=0, n==0, sumdiv(n, d, eulerphi(n/d) * C(3*d-1,d)) / (2*n) ); */
/* Joerg Arndt, Oct 21 2012 */

Formula

From Joerg Arndt, Oct 21 2012: (Start)

a(n) = sum( d divides n, phi(n/d) * C(3*d,d) ) / (3*n) for n>=1, a(0)=1.

a(n) = sum( d divides n, phi(n/d) * C(3*d-1,d) ) / (2*n) for n>=1, a(0)=1.

a(n) = A047996(3*n,n).

(End)

a(n) ~ 3^(3*n) / (2^(2*n+1) * sqrt(3*Pi) * n^(3/2)). - Vaclav Kotesovec, Aug 22 2015

Extensions

a(0)=1 prepended by Joerg Arndt, Oct 21 2012