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

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

Offset

3,1

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

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

Mathematica

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

Prog

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

Crossrefs

Sequence in context: A296050 A347666 A055882 * A319949 A304070 A349105

Adjacent sequences: A002298 A002299 A002300 * A002302 A002303 A002304

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved