cp's OEIS Frontend

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

%I A121774 #26 May 22 2024 02:12:57
%S A121774 1,1,2,6,33,260,2812,37450,597965,11111134,235796238,5628851294,
%T A121774 149346730841,4361070182716,139013934267864,4803839602537336,
%U A121774 178901440745010273,7143501829211426576,304465936544543927890,13797052631578947368422,662424832016591020302673,33591880889828764020700500
%N A121774 Number of n-bead necklaces with n+1 colors, divided by (n+1), for n>0, with a(0)=1.
%H A121774 Stefano Spezia, <a href="/A121774/b121774.txt">Table of n, a(n) for n = 0..350</a>
%H A121774 Darij Grinberg and Peter Mao, <a href="https://arxiv.org/abs/2405.08937">Necklaces over a group with identity product</a>, arXiv:2405.08937 [math.CO], 2024. See p. 23.
%F A121774 a(n) = (1/n)*Sum_{d|n} phi(n/d)*(n+1)^(d-1), for n>0, with a(0)=1.
%F A121774 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
%F A121774 a(n) = A121773(n)/(n+1). - _Amiram Eldar_, Aug 15 2023
%t A121774 a[n_] := DivisorSum[n, EulerPhi[n/#] * (n+1)^(#-1) &] / n; a[0] = 1; Array[a, 20, 0] (* _Amiram Eldar_, Aug 15 2023 *)
%o A121774 (PARI) a(n)=if(n==0,1,(1/n)*sumdiv(n,d,eulerphi(n/d)*(n+1)^(d-1)))
%o A121774 (PARI) /* a(n) = Sum_{k=0..[n/2]} A152290(n, n*k):  */
%o A121774 {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)}
%o A121774 {a(n)=sum(k=0,n\2,A152290(n, n*k))}
%o A121774 for(n=0,20,print1(a(n),", ")) \\ _Paul D. Hanna_, Jul 18 2013
%Y A121774 Cf. A000010, A121773, A056665, A152290.
%K A121774 nonn
%O A121774 0,3
%A A121774 _Paul D. Hanna_, Aug 20 2006