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 A001470 Number of degree-n permutations of order dividing 3. (Formerly M2782 N1118) 44

%I M2782 N1118

%S 1,1,1,3,9,21,81,351,1233,5769,31041,142011,776601,4874013,27027729,

%T 168369111,1191911841,7678566801,53474964993,418199988339,

%U 3044269834281,23364756531621,199008751634001,1605461415071823

%N Number of degree-n permutations of order dividing 3.

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

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

%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 T. D. Noe, <a href="/A001470/b001470.txt">Table of n, a(n) for n=0..100</a>

%F 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)

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

%F a(n)= a(n-1)+(n-1)*(n-2)*a(n-3) [From _Geoffrey Critzer_, Feb 03 2009]

%F 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 _Vladimir Kruchinin_, Sep 07 2010]

%t 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 *)

%o (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 _Vladimir Kruchinin_, Sep 07 2010]

%Y Cf. A000085, A001472.

%Y Column k=3 of A008307.

%K easy,nonn,nice

%O 0,4

%A _N. J. A. Sloane_, J. H. Conway and _Simon Plouffe_

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Last modified May 25 08:17 EDT 2013. Contains 225646 sequences.