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

%I A215716 #25 Oct 21 2016 07:49:09
%S A215716 1,1,2,6,24,96,576,4032,32256,290304,2612736,28740096,344881152,
%T A215716 4483454976,62768369664,878757175296,14060114804736,239021951680512,
%U A215716 4302395130249216,81745507474735104,1553164642019966976,32616457482419306496,717562064613224742912
%N A215716 Number of permutations on n points admitting a fifth root.
%C A215716 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.
%H A215716 Eric M. Schmidt, <a href="/A215716/b215716.txt">Table of n, a(n) for n = 0..200</a>
%H A215716 H. S. Wilf, <a href="http://www.math.upenn.edu/~wilf/DownldGF.html">Generatingfunctionology</a>, 2nd edn., Academic Press, NY, 1994, p. 148-149, Thms. 4.8.2 and 4.8.3.
%F A215716 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! + ... .
%p A215716 with(combinat):
%p A215716 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A215716       add(`if`(irem(j, igcd(i, 5))<>0, 0, (i-1)!^j*
%p A215716       multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1)), j=0..n/i)))
%p A215716     end:
%p A215716 a:= n-> b(n$2):
%p A215716 seq(a(n), n=0..25);  # _Alois P. Heinz_, Sep 08 2014
%t A215716 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_ *)
%o A215716 (PARI)
%o A215716 { 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)))); }
%o A215716 { exp5(y, prec) = subst(serconvol(exp(x + O(x^prec)), 1/(1-x^5) + O(x^prec)), x, y); }
%Y A215716 Cf. A003483, A103619, A103620, A215717, A215718.
%Y A215716 Column k=5 of A247005.
%K A215716 nonn
%O A215716 0,3
%A A215716 _Eric M. Schmidt_, Aug 22 2012