A053502 Number of degree-n permutations of order dividing 12.

1, 1, 2, 6, 24, 96, 576, 3312, 21456, 152784, 1237536, 9984096, 133494912, 1412107776, 16369357824, 206123325696, 2866280276736, 36809077162752, 592066290710016, 8800038127378944, 136876273991755776, 2197453620220010496, 37915306084793106432

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^4/4 + x^6/6 + x^12/12).

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, 4, 6, 12])))
    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, 4, 6, 12}}]]]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Apr 24 2014, after Alois P. Heinz *)
With[{m = 30}, CoefficientList[Series[Exp[x +x^2/2 +x^3/3 +x^4/4 +x^6/6 + x^12/12], {x, 0, m}], x]*Range[0, m]!] (* G. C. Greubel, May 15 2019 *)

Prog

(PARI) my(x='x+O('x^30)); Vec(serlaplace( exp(x + x^2/2 + x^3/3 + x^4/4 + x^6/6 + x^12/12) )) \\ G. C. Greubel, May 15 2019
(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x + x^2/2 + x^3/3 + x^4/4 + x^6/6 + x^12/12) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 15 2019
(Sage) m = 30; T = taylor(exp(x + x^2/2 + x^3/3 + x^4/4 + x^6/6 + x^12/12), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, May 15 2019

Crossrefs

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

Column k=12 of A008307.

Sequence in context: A152331 A152324 A147887 * A053504 A215716 A060725

Adjacent sequences: A053499 A053500 A053501 * A053503 A053504 A053505

Keyword

nonn

Author

N. J. A. Sloane, Jan 15 2000

Status

approved