A005388 - OEIS

%I M1293 #45 Jul 06 2024 14:03:35

%S 1,1,2,4,16,56,256,1072,11264,78976,672256,4653056,49810432,433429504,

%T 4448608256,39221579776,1914926104576,29475151020032,501759779405824,

%U 6238907914387456,120652091860975616,1751735807564578816,29062253310781161472,398033706586943258624

%N Number of degree-n permutations of order a power of 2.

%C Differs from A053503 first at n=32. - _Alois P. Heinz_, Feb 14 2013

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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

%H Alois P. Heinz, <a href="/A005388/b005388.txt">Table of n, a(n) for n = 0..200</a>

%H J. M. Møller, <a href="http://arxiv.org/abs/1502.01317">Euler characteristics of equivariant subcategories</a>, arXiv preprint arXiv:1502.01317, 2015. See page 20.

%H L. Moser and M. Wyman, <a href="http://dx.doi.org/10.4153/CJM-1955-020-0">On solutions of x^d = 1 in symmetric groups</a>, Canad. J. Math., 7 (1955), 159-168.

%H A. Recski, <a href="/A005387/a005387_1.pdf">Enumerating partitional matroids</a>, Preprint.

%H A. Recski & N. J. A. Sloane, <a href="/A005387/a005387.pdf">Correspondence, 1975</a>

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

%F 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

%p a:= proc(n) option remember; `if`(n<0, 0, `if`(n=0, 1,

%p        add(mul(n-i, i=1..2^j-1)*a(n-2^j), j=0..ilog2(n))))

%p     end:

%p seq(a(n), n=0..25);  # _Alois P. Heinz_, Feb 14 2013

%t 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 *)

%o (Magma)

%o R<x>:=PowerSeriesRing(Rationals(), 40);

%o f:= func< x | Exp( (&+[x^(2^j)/2^j: j in [0..14]]) ) >;

%o Coefficients(R!(Laplace( f(x) ))); // _G. C. Greubel_, Nov 17 2022

%o (SageMath)

%o def f(x): return exp(sum(x^(2^j)/2^j for j in range(15)))

%o def A005388_list(prec):

%o     P.<x> = PowerSeriesRing(QQ, prec)

%o     return P( f(x) ).egf_to_ogf().list()

%o A005388_list(40) # _G. C. Greubel_, Nov 17 2022

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

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

%K nonn,nice,easy

%O 0,3

%A _N. J. A. Sloane_ and _J. H. Conway_