A002301 a(n) = n! / 3.

2, 8, 40, 240, 1680, 13440, 120960, 1209600, 13305600, 159667200, 2075673600, 29059430400, 435891456000, 6974263296000, 118562476032000, 2134124568576000, 40548366802944000, 810967336058880000, 17030314057236480000, 374666909259202560000

Offset: 3

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

a(n) is the number of n-permutations having 1, 2 and 3 in the same cycle. - Geoffrey Critzer, Apr 26 2009

a(n) is the total number of 3-cycles in all n-permutations. - N. J. A. Sloane, Jul 22 2009

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

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

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

Links

• N. J. A. Sloane, Table of n, a(n) for n = 3..30
• M. A. A., The 67th William Lowell Putnam Mathematical Competition Problem A4.
• L. Solus, Local h*-polynomials of some weighted projective spaces, arXiv:1807.08223 [math.CO], 2018.
• Index to divisibility sequences
• Index entries for sequences related to factorial numbers

Programs

Mathematica

f[n_]:=n!/3;Array[f,4!,3] (* Vladimir Joseph Stephan Orlovsky, Oct 21 2009 *)

PARI

a(n)=n!/3 \\ Charles R Greathouse IV, Jan 12 2012

Formula

E.g.f. with offset = 0: 2/((1-x)^4). - Sergei N. Gladkovskii, Aug 16 2012

E.g.f.: x^3/(3*(1-x)). - Geoffrey Critzer, Aug 26 2012

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