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A215717 Number of permutations on n points admitting a sixth root. 7
1, 1, 1, 1, 4, 40, 190, 1330, 8680, 52920, 340200, 6237000, 76211520, 1098857760, 11677585920, 109679169600, 1497396700800, 41977644508800, 783593969558400, 15973899557616000, 263524120417958400, 3733362595368806400, 64262934423790502400 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

Cf. A003483, A103619, A103620, A215716, A215718.

Column k=6 of A247005.

Sequence in context: A238328 A009355 A061132 * A270099 A271274 A271286

Adjacent sequences:  A215714 A215715 A215716 * A215718 A215719 A215720

KEYWORD

nonn

AUTHOR

Eric M. Schmidt, Aug 23 2012

STATUS

approved

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Last modified July 18 15:44 EDT 2019. Contains 325144 sequences. (Running on oeis4.)