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A001470
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Number of degree-n permutations of order dividing 3.
(Formerly M2782 N1118)
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43
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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; internal format)
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OFFSET
| 0,4
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REFERENCES
| L. Moser and M. Wyman, On solutions of x^d = 1 in symmetric groups, Canad. J. Math., 7 (1955), 159-168.
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.
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..100
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FORMULA
| a(n) = Sum from j=0 to Int(n/3) of n!/(j!*(n-3j)!*(3^j)) (the latter formula from Roger CUCULIERE, cuculier(AT)sophocle.imaginet.fr)
E.g.f.: exp(x+1/3*x^3).
a(n)= a(n-1)+(n-1)*(n-2)*a(n-3) [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), 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. [From Kruchinin Vladimir (kru(AT)ie.tusur.ru), Sep 07 2010]
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MATHEMATICA
| a[n_] := HypergeometricPFQ[{(1-n)/3, (2-n)/3, -n/3}, {}, -9]; Table[a[n], {n, 0, 23}] (* From Jean-François Alcover, Nov 03 2011 *)
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PROG
| (Other) 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); (for Maxima) [From Kruchinin Vladimir (kru(AT)ie.tusur.ru), Sep 07 2010]
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CROSSREFS
| Cf. A000085, A001472.
Sequence in context: A004667 A073947 A062811 * A118932 A053499 A146909
Adjacent sequences: A001467 A001468 A001469 * A001471 A001472 A001473
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KEYWORD
| easy,nonn,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), J. H. Conway and Simon Plouffe (simon.plouffe(AT)gmail.com)
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