A121774 Number of n-bead necklaces with n+1 colors, divided by (n+1), for n>0, with a(0)=1.

1, 1, 2, 6, 33, 260, 2812, 37450, 597965, 11111134, 235796238, 5628851294, 149346730841, 4361070182716, 139013934267864, 4803839602537336, 178901440745010273, 7143501829211426576, 304465936544543927890, 13797052631578947368422, 662424832016591020302673, 33591880889828764020700500

Offset: 0

Author

Paul D. Hanna, Aug 20 2006

Keywords

• nonn

Links

• Stefano Spezia, Table of n, a(n) for n = 0..350
• Darij Grinberg and Peter Mao, Necklaces over a group with identity product, arXiv:2405.08937 [math.CO], 2024. See p. 23.

Crossrefs

Cf. A000010, A121773, A056665, A152290.

Programs

Mathematica

a[n_] := DivisorSum[n, EulerPhi[n/#] * (n+1)^(#-1) &] / n; a[0] = 1; Array[a, 20, 0] (* Amiram Eldar, Aug 15 2023 *)

PARI

a(n)=if(n==0,1,(1/n)*sumdiv(n,d,eulerphi(n/d)*(n+1)^(d-1)))

PARI

/* a(n) = Sum_{k=0..[n/2]} A152290(n, n*k):  */
{A152290(n,k)=local(e_q=1+sum(j=1,n,x^j/prod(i=1,j,(q^i-1)/(q-1))),LW_q=serreverse(x/e_q+x^2*O(x^n))/x); polcoeff(polcoeff(LW_q+x*O(x^n),n,x)*prod(i=1,n,(q^i-1)/(q-1))+q*O(q^k),k,q)}
{a(n)=sum(k=0,n\2,A152290(n, n*k))}
for(n=0,20,print1(a(n),", ")) \\ Paul D. Hanna, Jul 18 2013

Formula

a(n) = (1/n)*Sum_{d|n} phi(n/d)*(n+1)^(d-1), for n>0, with a(0)=1.

a(n) = Sum_{k=0..[n/2]} A152290(n, n*k), where A152290 is a triangle of coefficients in a q-analog of the LambertW function. - Paul D. Hanna, Jul 18 2013

a(n) = A121773(n)/(n+1). - Amiram Eldar, Aug 15 2023