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A215717 Number of permutations on n points admitting a sixth root.

%I A215717 #16 Oct 21 2016 06:22:41
%S A215717 1,1,1,1,4,40,190,1330,8680,52920,340200,6237000,76211520,1098857760,
%T A215717 11677585920,109679169600,1497396700800,41977644508800,
%U A215717 783593969558400,15973899557616000,263524120417958400,3733362595368806400,64262934423790502400
%N A215717 Number of permutations on n points admitting a sixth root.
%C A215717 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).
%H A215717 Eric M. Schmidt, <a href="/A215717/b215717.txt">Table of n, a(n) for n = 0..200</a>
%H A215717 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 A215717 E.g.f.: prod(m>=1, E_(gcd(m,6))(x^m/m) ), where E_j(x) = 1 + x^j/j! + x^(2j)/(2j)! + ... .
%p A215717 with(combinat):
%p A215717 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A215717       add(`if`(irem(j, igcd(i, 6))<>0, 0, (i-1)!^j*
%p A215717       multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1)), j=0..n/i)))
%p A215717     end:
%p A215717 a:= n-> b(n$2):
%p A215717 seq(a(n), n=0..25);  # _Alois P. Heinz_, Sep 08 2014
%t A215717 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_ *)
%o A215717 (PARI)
%o A215717 { A215717_list(numterms) = Vec(serlaplace(prod(m=1, numterms, expthin(gcd(m, 6), x^m/m, numterms\m+1))) + O(x^numterms)); }
%o A215717 { expthin(j, y, prec) = subst(serconvol(exp(x + O(x^prec)), 1/(1-x^j) + O(x^prec)), x, y); }
%Y A215717 Cf. A003483, A103619, A103620, A215716, A215718.
%Y A215717 Column k=6 of A247005.
%K A215717 nonn
%O A215717 0,5
%A A215717 _Eric M. Schmidt_, Aug 23 2012