A053496 Number of degree-n permutations of order dividing 6.

1, 1, 2, 6, 18, 66, 396, 2052, 12636, 91548, 625176, 4673736, 43575192, 377205336, 3624289488, 38829340656, 397695226896, 4338579616272, 54018173703456, 641634784488288, 8208962893594656, 113809776294348576, 1526808627197721792, 21533423236302943296

Offset

0,3

References

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.10.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..200

L. Moser and M. Wyman, On solutions of x^d = 1 in symmetric groups, Canad. J. Math., 7 (1955), 159-168.

Formula

E.g.f.: exp(x +x^2/2 +x^3/3 +x^6/6).

D-finite with recurrence a(n) -a(n-1) +(-n+1)*a(n-2) -(n-1)*(n-2)*a(n-3) -(n-5)*(n-1)*(n-2)*(n-3)*(n-4)*a(n-6)=0. - R. J. Mathar, Jul 04 2023

Maple

a:= proc(n) option remember; `if`(n<0, 0, `if`(n=0, 1,
       add(mul(n-i, i=1..j-1)*a(n-j), j=[1, 2, 3, 6])))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Feb 14 2013

Mathematica

a[n_] := a[n] = If[n<0, 0, If[n == 0, 1, Sum[Product[n-i, {i, 1, j-1}]*a[n-j], {j, {1, 2, 3, 6}}]]]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Mar 03 2014, after Alois P. Heinz *)
With[{m = 30}, CoefficientList[Series[Exp[x +x^2/2 +x^3/3 +x^6/6], {x, 0, m}], x]*Range[0, m]!] (* G. C. Greubel, May 14 2019 *)

Prog

(PARI) my(x='x+O('x^30)); Vec(serlaplace( exp(x+x^2/2+x^3/3+x^6/6) )) \\ G. C. Greubel, May 14 2019
(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x + x^2/2 +x^3/3 +x^6/6) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 14 2019
(Sage) m = 30; T = taylor(exp(x +x^2/2 +x^3/3 +x^6/6), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, May 14 2019

Crossrefs

Cf. A000085, A001470, A001472, A053495-A053505, A005388, A261317.

Column k=6 of A008307.

Sequence in context: A150077 A173385 A057693 * A079577 A150078 A150079

Adjacent sequences: A053493 A053494 A053495 * A053497 A053498 A053499

Keyword

nonn

Author

N. J. A. Sloane, Jan 15 2000

Status

approved