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A001470 Number of degree-n permutations of order dividing 3.
(Formerly M2782 N1118)
44
1, 1, 1, 3, 9, 21, 81, 351, 1233, 5769, 31041, 142011, 776601, 4874013, 27027729, 168369111, 1191911841, 7678566801, 53474964993, 418199988339, 3044269834281, 23364756531621, 199008751634001, 1605461415071823 (list; graph; refs; listen; history; text; internal format)
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

T. D. Noe, Table of n, a(n) for n=0..100

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

Marcello Artioli, Giuseppe Dattoli, Silvia Licciardi, Simonetta Pagnutti, Motzkin Numbers: an Operational Point of View, arXiv:1703.07262 [math.CO], 2017. See p. 27.

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 from j=0 to Int(n/3) of n!/(j!*(n-3j)!*(3^j)) (the latter formula from Roger Cuculière).

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

a(n) = a(n-1)+(n-1)*(n-2)*a(n-3). - Geoffrey Critzer, Feb 03 2009

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), n>0. - 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+1/3 x^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 */

CROSSREFS

Cf. A000085, A001472.

Column k=3 of A008307.

Sequence in context: A073947 A062811 A236856 * 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

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Last modified July 23 14:59 EDT 2017. Contains 289688 sequences.