A215718 Number of permutations on n points admitting a seventh root.

1, 1, 2, 6, 24, 120, 720, 4320, 34560, 311040, 3110400, 34214400, 410572800, 5337446400, 69386803200, 1040802048000, 16652832768000, 283098157056000, 5095766827008000, 96819569713152000, 1936391394263040000, 38727827885260800000, 852012213475737600000

Offset

0,3

Comments

a(n) is the number of permutations of n points such that for all positive m, the number of (7m)-cycles is a multiple of 7.

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.: (1 - x^7)^(1/7)/(1 - x) * Product_{m>=1} E_7(x^(7m)/(7m)) ), where E_7(x) = 1 + x^7/7! + x^14/14! + ... .

Maple

with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(`if`(irem(j, igcd(i, 7))<>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, 7]] != 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)
{ A215718_list(numterms) = Vec(serlaplace((1 - x^7 + O(x^numterms))^(1/7)/(1-x) * prod(m=1, numterms\7, exp7(x^(7*m)/(7*m), numterms\(7*m)+1)))); }
{ exp7(y, prec) = subst(serconvol(exp(x + O(x^prec)), 1/(1-x^7) + O(x^prec)), x, y); }

Crossrefs

Cf. A003483, A103619, A103620, A215716, A215717.

Column k=7 of A247005.

Sequence in context: A179365 A070947 A267386 * A060727 A152350 A152368

Adjacent sequences: A215715 A215716 A215717 * A215719 A215720 A215721

Keyword

nonn

Author

Eric M. Schmidt, Aug 23 2012

Status

approved