A053500 Number of degree-n permutations of order dividing 10.

1, 1, 2, 4, 10, 50, 220, 1240, 6140, 32860, 602200, 5668400, 62030200, 522328600, 4487190800, 62591332000, 715163146000, 9573774122000, 105731659828000, 1187355279592000, 29205778751300000, 481597207656340000, 9086318388933400000, 132525988426667120000

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^5/5 + x^10/10).

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, 5, 10])))
    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, 5, 10}}]]]; 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^5/5 +x^10/10], {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^5/5 + x^10/10) )) \\ G. C. Greubel, May 15 2019
(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x + x^2/2 + x^5/5 + x^10/10) )); [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^5/5 + x^10/10), 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=10 of A008307.

Sequence in context: A173488 A000613 A322294 * A214724 A326325 A080090

Adjacent sequences: A053497 A053498 A053499 * A053501 A053502 A053503

Keyword

nonn

Author

N. J. A. Sloane, Jan 15 2000

Status

approved