A053503 Number of degree-n permutations of order dividing 16.

1, 1, 2, 4, 16, 56, 256, 1072, 11264, 78976, 672256, 4653056, 49810432, 433429504, 4448608256, 39221579776, 1914926104576, 29475151020032, 501759779405824, 6238907914387456, 120652091860975616, 1751735807564578816, 29062253310781161472, 398033706586943258624

Offset

0,3

Comments

Differs from A005388 first at n=32. - Alois P. Heinz, Feb 14 2013

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^4/4 + x^8/8 + x^16/16).

Maple

a:= proc(n) option remember; `if`(n<0, 0, `if`(n=0, 1,
       add(mul(n-i, i=1..2^j-1)*a(n-2^j), j=0..4)))
    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, 2^j-1}]* a[n-2^j], {j, 0, 4}]]]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Mar 19 2014, after Alois P. Heinz *)
With[{m = 30}, CoefficientList[Series[Exp[x +x^2/2 +x^4/4 +x^8/8 + x^16/16], {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^4/4 + x^8/8 + x^16/16) )) \\ G. C. Greubel, May 15 2019
(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x + x^2/2 + x^4/4 + x^8/8 + x^16/16) )); [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^4/4 + x^8/8 + x^16/16), 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=16 of A008307.

Sequence in context: A001472 A053498 A005388 * A308381 A153957 A068789

Adjacent sequences: A053500 A053501 A053502 * A053504 A053505 A053506

Keyword

nonn

Author

N. J. A. Sloane, Jan 15 2000

Status

approved