A001470 Number of degree-n permutations of order dividing 3. (Formerly M2782 N1118)

1, 1, 1, 3, 9, 21, 81, 351, 1233, 5769, 31041, 142011, 776601, 4874013, 27027729, 168369111, 1191911841, 7678566801, 53474964993, 418199988339, 3044269834281, 23364756531621, 199008751634001, 1605461415071823

Offset

0,4

References

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

Seiichi Manyama, Table of n, a(n) for n = 0..631 (terms 0..100 from T. D. Noe)

Joerg Arndt, Generating Random Permutations, PhD thesis, Australian National University, Canberra, Australia, (2010).

Marcello Artioli, Giuseppe Dattoli, Silvia Licciardi, and Simonetta Pagnutti, Motzkin Numbers: an Operational Point of View, arXiv:1703.07262 [math.CO], 2017. See p. 7.

L. Moser and M. Wyman, On solutions of x^d = 1 in symmetric groups, Canad. J. Math., 7 (1955), 159-168.

Formula

a(n) = Sum_{j=0..floor(n/3)} n!/(j!*(n-3j)!*(3^j)) (the latter formula from Roger Cuculière).

E.g.f.: exp(x + (1/3)*x^3).

D-finite with recurrence: a(n) = a(n-1) + (n-1)*(n-2)*a(n-3). - Geoffrey Critzer, Feb 03 2009

a(n) = n!*Sum_{k=floor(n/3)..n, n - k == 0 (mod 2)} binomial(k,(3*k-n)/2)*(1/3)^((n-k)/2)/k!. - Vladimir Kruchinin, Sep 07 2010

a(n) ~ n^(2*n/3)*exp(n^(1/3)-2*n/3)/sqrt(3) * (1 - 1/(6*n^(1/3)) + 25/(72*n^(2/3))). - Vaclav Kotesovec, Jul 28 2013

Mathematica

a[n_] := HypergeometricPFQ[{(1-n)/3, (2-n)/3, -n/3}, {}, -9]; Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Nov 03 2011 *)
With[{nn=30}, CoefficientList[Series[Exp[x+x^3/3], {x, 0, nn}], x] Range[ 0, nn]!] (* Harvey P. Dale, Aug 12 2016 *)

Prog

(Maxima) a(n):=n!*sum(if mod(n-k, 2)=0 then binomial(k, (3*k-n)/2)*(1/3)^((n-k)/2)/k! else 0, k, floor(n/3), n); /* Vladimir Kruchinin, Sep 07 2010 */
(Magma) R<x>:=PowerSeriesRing(Rationals(), 40); Coefficients(R!(Laplace( Exp(x+x^3/3) ))); // G. C. Greubel, Sep 03 2023
(SageMath)
def A001470_list(prec):
    P.<x> = PowerSeriesRing(QQ, prec)
    return P( exp(x+x^3/3) ).egf_to_ogf().list()
A001470_list(40) # G. C. Greubel, Sep 03 2023

Crossrefs

Cf. A000085, A001472.

Column k=3 of A008307.

Sequence in context: A236856 A338534 A318843 * A118932 A053499 A218003

Adjacent sequences: A001467 A001468 A001469 * A001471 A001472 A001473

Keyword

easy,nonn,nice

Author

N. J. A. Sloane, J. H. Conway and Simon Plouffe

Status

approved