|
| |
|
|
A121774
|
|
Number of n-bead necklaces with n+1 colors, divided by (n+1), for n>0, with a(0)=1.
|
|
3
|
|
|
|
1, 1, 2, 6, 33, 260, 2812, 37450, 597965, 11111134, 235796238, 5628851294, 149346730841, 4361070182716, 139013934267864, 4803839602537336, 178901440745010273, 7143501829211426576, 304465936544543927890, 13797052631578947368422, 662424832016591020302673, 33591880889828764020700500
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,3
|
|
|
LINKS
|
|
|
|
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
|
|
|
MATHEMATICA
|
a[n_] := DivisorSum[n, EulerPhi[n/#] * (n+1)^(#-1) &] / n; a[0] = 1; Array[a, 20, 0] (* Amiram Eldar, Aug 15 2023 *)
|
|
|
PROG
|
(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))}
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
