OFFSET
0,5
COMMENTS
a(n) is the number of permutations of n points such that for all positive m, the number of m-cycles is a multiple of gcd(m, 6).
LINKS
Eric M. Schmidt, Table of n, a(n) for n = 0..200
H. S. Wilf, Generatingfunctionology, 2nd edn., Academic Press, NY, 1994, p. 148-149, Thms. 4.8.2 and 4.8.3.
FORMULA
E.g.f.: prod(m>=1, E_(gcd(m,6))(x^m/m) ), where E_j(x) = 1 + x^j/j! + x^(2j)/(2j)! + ... .
MAPLE
with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(`if`(irem(j, igcd(i, 6))<>0, 0, (i-1)!^j*
multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1)), j=0..n/i)))
end:
a:= n-> b(n$2):
seq(a(n), n=0..25); # Alois P. Heinz, Sep 08 2014
MATHEMATICA
multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[If[Mod[j, GCD[i, 6]] != 0, 0, (i-1)!^j*multinomial[n, Prepend[Table[i, {j}], n-i*j]]/j!*b[n-i*j, i - 1]], {j, 0, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Oct 21 2016, after Alois P. Heinz *)
PROG
(PARI)
{ A215717_list(numterms) = Vec(serlaplace(prod(m=1, numterms, expthin(gcd(m, 6), x^m/m, numterms\m+1))) + O(x^numterms)); }
{ expthin(j, y, prec) = subst(serconvol(exp(x + O(x^prec)), 1/(1-x^j) + O(x^prec)), x, y); }
CROSSREFS
KEYWORD
nonn
AUTHOR
Eric M. Schmidt, Aug 23 2012
STATUS
approved