A005388 Number of degree-n permutations of order a power of 2.

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

Author

N. J. A. Sloane and J. H. Conway

Keywords

• nonn
• nice
• easy

Comments

Differs from A053503 first at n=32. - Alois P. Heinz, Feb 14 2013

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• 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
• J. M. Møller, Euler characteristics of equivariant subcategories, arXiv preprint arXiv:1502.01317, 2015. See page 20.
• L. Moser and M. Wyman, On solutions of x^d = 1 in symmetric groups, Canad. J. Math., 7 (1955), 159-168.
• A. Recski, Enumerating partitional matroids, Preprint.
• A. Recski & N. J. A. Sloane, Correspondence, 1975

Crossrefs

Cf. A000085, A001470, A001472, A053495, A053496, A053497, A053498, A053499.

Cf. A053500, A053501, A053502, A053503, A053504, A053505.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 40);
f:= func< x | Exp( (&+[x^(2^j)/2^j: j in [0..14]]) ) >;
Coefficients(R!(Laplace( f(x) ))); // G. C. Greubel, Nov 17 2022

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..ilog2(n))))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Feb 14 2013

Mathematica

max = 23; CoefficientList[ Series[ Exp[ Sum[x^2^m/2^m, {m, 0, max}]], {x, 0, max}], x]*Range[0, max]! (* Jean-François Alcover, Sep 10 2013 *)

SageMath

def f(x): return exp(sum(x^(2^j)/2^j for j in range(15)))
def A005388_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(x) ).egf_to_ogf().list()
A005388_list(40) # G. C. Greubel, Nov 17 2022

Formula

E.g.f.: exp(Sum_{m>=0} x^(2^m)/2^m).

E.g.f.: 1/Product_{k>=1} (1 - x^(2*k-1))^(mu(2*k-1)/(2*k-1)), where mu() is the Moebius function. - Seiichi Manyama, Jul 06 2024