A072605 Number of necklaces with n beads over an n-ary alphabet {a1,a2,...,an} such that #(w,a1) >= #(w,a2) >= ... >= #(w,ak) >= 0, where #(w,x) counts the letters x in word w.

1, 1, 2, 4, 13, 50, 270, 1641, 11945, 96784, 887982, 8939051, 99298354, 1195617443, 15619182139, 219049941201, 3293800835940, 52746930894774, 897802366250126, 16167544246362567, 307372573011579188, 6148811682561390279, 129164845357784003661

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..200

Frank Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.

F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]

Eric Weisstein's world of Mathematics, Necklaces

Formula

a(n) = (1/n) * Sum_{d|n} phi(n/d) * A005651(d) for n > 0. - Andrew Howroyd, Sep 25 2017

See Mathematica line.

a(n) ~ c * (n-1)!, where c = Product_{k>=2} 1/(1-1/k!) = A247551 = 2.52947747207915264818011615... . - Vaclav Kotesovec, Aug 27 2015

Mathematica

neck[li:{__Integer}] := Module[{n, d}, n=Plus@@li; d=n-First[li]; Fold[ #1+(EulerPhi[ #2]*(n/#2)!)/Times@@((li/#2)!)&, 0, Divisors[GCD@@li]]/n]; Table[ Plus@@(neck /@ IntegerPartitions[n]), {n, 24}]

Prog

(PARI) a(n)={if(n==0, 1, my(p=prod(k=1, n, 1/(1-x^k/k!) + O(x*x^n))); sumdiv(n, d, eulerphi(n/d)*d!*polcoeff(p, d))/n)} \\ Andrew Howroyd, Dec 20 2017

Crossrefs

Cf. A005651, A052307, A247551, A261531, A261599, A261600.

Sequence in context: A058134 A246012 A069730 * A330344 A389662 A161905

Adjacent sequences: A072602 A072603 A072604 * A072606 A072607 A072608

Keyword

easy,nonn

Author

Wouter Meeussen, Aug 06 2002

Extensions

a(0)=1 prepended by Alois P. Heinz, Aug 23 2015

Name changed by Andrew Howroyd, Sep 25 2017

Status

approved