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A002301 a(n) = n! / 3.
(Formerly M1861 N0737)

%I M1861 N0737

%S 2,8,40,240,1680,13440,120960,1209600,13305600,159667200,2075673600,

%T 29059430400,435891456000,6974263296000,118562476032000,

%U 2134124568576000,40548366802944000,810967336058880000,17030314057236480000,374666909259202560000

%N a(n) = n! / 3.

%C a(n) is the number of n-permutations having 1, 2 and 3 in the same cycle. - _Geoffrey Critzer_, Apr 26 2009

%C a(n) is the total number of 3-cycles in all n-permutations. - _N. J. A. Sloane_, Jul 22 2009

%C a(n+1) is the number of local maxima summed over all partitions of length n where n>1. - _Michael Somos_, Jul 19 2012

%C For n>2, n!/3 is the number of lattice points in the open parallelepiped of the factoradic n-simplex. See Remark 3.1 in the article by L. Solus below. - _Liam Solus_, Aug 23 2018

%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 Letterio Toscano, Sulla Derivata di Ordinen della Funzione tg(x), Tohoku Math. J., 42 (1936), 144-154.

%H N. J. A. Sloane, <a href="/A002301/b002301.txt">Table of n, a(n) for n = 3..30</a>

%H M. A. A., <a href="http://math.hawaii.edu/home/pdf/putnam/2006.pdf">The 67th William Lowell Putnam Mathematical Competition</a> Problem A4.

%H L. Solus, <a href="https://arxiv.org/abs/1807.08223">Local h*-polynomials of some weighted projective spaces</a>, arXiv:1807.08223 [math.CO], 2018.

%H <a href="/index/Di#divseq">Index to divisibility sequences</a>

%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>

%F E.g.f. with offset = 0: 2/((1-x)^4). - _Sergei N. Gladkovskii_, Aug 16 2012

%F E.g.f.: x^3/(3*(1-x)). - _Geoffrey Critzer_, Aug 26 2012

%F G.f. 2 + 8*x/(G(0)-4*x) where G(k) = x*(k+4) + 1 - x*(k+5)/G(k+1); (continued fraction, Euler's 1st kind, 1-step). - _Sergei N. Gladkovskii_, Aug 15 2012

%t f[n_]:=n!/3;Array[f,4!,3] (* _Vladimir Joseph Stephan Orlovsky_, Oct 21 2009 *)

%o (PARI) a(n)=n!/3 \\ _Charles R Greathouse IV_, Jan 12 2012

%K nonn,easy

%O 3,1

%A _N. J. A. Sloane_

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Last modified October 14 09:57 EDT 2019. Contains 327995 sequences. (Running on oeis4.)