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A082936 a(n) = (1/(3*n))*Sum_{d|n, d even} phi(2*n/d)*binomial(3d/2,d). 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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.

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

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-Fran├žois Alcover, Dec 02 2015 *)

PROG

(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 */

CROSSREFS

Cf. A003239.

Column k=2 of A261494.

Sequence in context: A203266 A151084 A151085 * A205487 A323667 A030935

Adjacent sequences:  A082933 A082934 A082935 * A082937 A082938 A082939

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, May 26 2003

EXTENSIONS

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

STATUS

approved

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Last modified October 14 17:43 EDT 2019. Contains 328022 sequences. (Running on oeis4.)