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 A001470 Number of degree-n permutations of order dividing 3. (Formerly M2782 N1118) 45
 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 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, 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). 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} (if mod(n-k,2)=0 then binomial(k,(3*k-n)/2)*(1/3)^((n-k)/2)/k! else 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: A062811 A236856 A318843 * A118932 A053499 A218003 Adjacent sequences:  A001467 A001468 A001469 * A001471 A001472 A001473 KEYWORD easy,nonn,nice AUTHOR STATUS approved

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Last modified November 17 18:55 EST 2018. Contains 317276 sequences. (Running on oeis4.)