|
|
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.
|
|
8
|
|
|
1, 1, 2, 4, 13, 50, 270, 1641, 11945, 96784, 887982, 8939051, 99298354, 1195617443, 15619182139, 219049941201, 3293800835940, 52746930894774, 897802366250126, 16167544246362567, 307372573011579188, 6148811682561390279, 129164845357784003661
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,3
|
|
LINKS
|
Eric Weisstein's world of Mathematics, Necklaces
|
|
FORMULA
|
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
|
|
|
KEYWORD
|
easy,nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|