1,3

Also the number of permutations with no global descents, as introduced by Aguiar and Sottile [Corollaries 6.3, 6.4 and Remark 6.5]

Also the dimensions of the homogeneous components of the space of primitive elements of the Malvenuto-Reutenauer Hopf algebra of permutations. This result, due to Poirier and Reutenauer [Theoreme 2.1] is stated in this form in the work of Aguiar and Sottile [Corollary 6.3] and also in the work of Duchamp, Hivert and Thibon [Section 3.3]

Related to number of subgroups of index n-1 in free group of rank 2 (i.e. maximal number of subgroups of index n-1 in any 2-generator group). See Problem 5.13(b) in Stanley's Enumerative Combinatorics, Vol. 2.

Left border of triangle A144107 = A003319, with row sums = n!. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 11 2008]

Hankel transform is A059332. Hankel transform of aerated sequence is A137704(n+1). [From Paul Barry (pbarry(AT)wit.ie), Oct 07 2008]

for every n, a(n+1) is also the moment of order n for the probability density function rho(x)=exp(x)/(Ei(1,-x)*(Ei(1,-x)+2*I*Pi)) on the interval 0..infinity, with Ei the exponentiel-integral function. [From Groux Roland (roland.groux(AT)orange.fr), Jan 16 2009]

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Marcelo Aguiar (Texas A&M University) and Frank Sottile (University of Massachusetts at Amherst). math.CO/0203282 Structure of the Malvenuto-Reutenauer Hopf algebra of permutations.

F. R. K. Chung and R. L. Graham, Primitive juggling sequences, preprint, 2006.

L. Comtet, Sur les coefficients de l'inverse de la series formelle Sum n! t^n, Comptes Rend. Acad. Sci. Paris, A 275 (1972), 569-572.

L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 84 (#25), 262 (#14) and 295 (#16).

J. D. Dixon, Asymptotics of generating the symmetric and alternating groups, Electron. J. Combin., Item R56 of Volume 12(1), 2005.

G. Duchamp (University of Rouen), F. Hivert and J.-Y. Thibon (University of Marne-la-Vallee). math.CO/0105065 Noncommutative symmetric functions VI: Free quasi-symmetric functions and related algebras.

I. M. Gessel and R. P. Stanley, Algebraic Enumeration, chapter 21 in Handbook of Combinatorics, Vol. 2, edited by R. L. Graham et al., The MIT Press, Mass, 1995.

A. King, Generating indecomposable permutations, Discrete Math., 306 (2006), 508-518.

P. Ossona de Mendez and P. Rosenstiehl, Transitivity and connectivity of permutations, Combinatorics, 24 (No. 3, 2004), 487-501.

L. Panaitopol, A formula for $\pi(x)$ applied to a result of Koninck-Ivi\'c, Nieuw Arch. Wisk. 5/1 55-56 (2000)

S. Poirier and C. Reutenauer, Algebres Hopf de tableaux, Ann. Sci. Math. Quebec 19 (95), no. 1, 79-90.

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.13(b).

R. P. Stanley, The Descent Set and Connectivity Set of a Permutation, Journal of Integer Sequences, Vol. 8 (2005), Article 05.3.8.

J. D. Dixon, The probability of generating the symmetric group, Math. Z. 110 (1969) 199-205.

T. D. Noe, Table of n, a(n) for n = 1..101

Joerg Arndt, Fxtbook

David Callan, Counting Stabilized-Interval-Free Permutations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.1.8.

I. M. Gessel and R. P. Stanley Algebraic Enumeration (See pages 7-8 for generating function.)

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 90

G.f.: 1-1/Sum (k! x^k ). Also a(n) = n! - Sum_{k=1..n-1} k!*a(n-k), n >= 1.

a(n) = (-1)^{n-1} * det {| 1! 2! ... n! | 1 1! ... (n-1)! | 0 1 1! ... (n-2)! | ... | 0 ... 0 1 1! |}

INVERTi transform of factorial numbers, A000142 starting from n=1. - Antti Karttunen (Antti.Karttunen(AT)iki.fi), May 30 2003

Gives the row sums of the triangle [0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...] DELTA [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, ...] where DELTA is the operator defined in A084938; this triangle, read by rows is the sequence : 1; 0, 1; 0, 1, 2; 0, 1, 6, 6; 0, 1, 12, 34, 24; 0, 1, 20, 110, 210, 120; 0, 1, 30, 270, 974, 1452, 720; ... - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Dec 30 2003

a(n+1)=Sum_{k,0<=k<=n}A089949(n,k). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 16 2006

L.g.f.: Sum_{n>=1} a(n)*x^n/n = log( Sum_{n>=0} n!*x^n ) . - Paul D. Hanna (pauldhanna(AT)juno.com), Sep 19 2007

G.f.: 1/(1-x/(1-2x/(1-2x/(1-3x/(1-3x/(1-4x/(1-4x/(1-.....))))))) (continued fraction); [From Paul Barry (pbarry(AT)wit.ie), Oct 07 2008]

For n > 0 let R be the n-th row of A090238. Then a(n) = Sum{i=0..n}(-1)^(i)*R[i]. [From Peter Luschny (peter(AT)luschny.de), Mar 13 2009]

INVERTi([seq(n!, n=1..20)]);

Leading diagonal of A059438.

Cf. A051296, A084938, A074664, A113869.

A144107 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 11 2008]

Sequence in context: A162326 A122455 A126390 this_sequence A158882 A000261 A111140

Adjacent sequences: A003316 A003317 A003318 this_sequence A003320 A003321 A003322

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).

More terms from Michael Somos, Jan 26 2000

Additional comments from Marcelo Aguiar (maguiar(AT)math.tamu.edu), Mar 28 2002