A215716 Number of permutations on n points admitting a fifth root.

1, 1, 2, 6, 24, 96, 576, 4032, 32256, 290304, 2612736, 28740096, 344881152, 4483454976, 62768369664, 878757175296, 14060114804736, 239021951680512, 4302395130249216, 81745507474735104, 1553164642019966976, 32616457482419306496, 717562064613224742912

Offset: 0

Author

Eric M. Schmidt, Aug 22 2012

Keywords

• nonn

Comments

a(n) is the number of permutations of n points such that for all positive m, the number of (5m)-cycles is a multiple of 5.

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.

Crossrefs

Cf. A003483, A103619, A103620, A215717, A215718.

Column k=5 of A247005.

Programs

Maple

with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(`if`(irem(j, igcd(i, 5))<>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, 5]] != 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 *)

PARI

{ A215716_list(numterms) = Vec(serlaplace((1 - x^5 + O(x^numterms))^(1/5)/(1-x) * prod(m=1, numterms\5, exp5(x^(5*m)/(5*m), numterms\(5*m)+1)))); }
{ exp5(y, prec) = subst(serconvol(exp(x + O(x^prec)), 1/(1-x^5) + O(x^prec)), x, y); }

Formula

E.g.f.: (1 - x^5)^(1/5)/(1 - x) * Product(E_5(x^(5m)/(5m)), m = 1 .. infinity), where E_5(x) = 1 + x^5/5! + x^10/10! + ... .