0,3

Number of {12,12*,1*2,21*}- and {12,12*,21,21*}-avoiding signed permutations in the hyperoctahedral group.

a(n) = number of permutations on [n] that avoid the patterns 2n1 and n12. An occurrence of a 2n1 pattern is a (scattered) subsequence a-n-b with a>b. - David Callan (callan(AT)stat.wisc.edu), Nov 29 2007

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

R. K. Guy, Unsolved Problems Number Theory, Section B44.

D. Kurepa, On the left factorial function !n. Math. Balkanica 1 1971 147-153.

T. D. Noe, Table of n, a(n) for n = 0..100

P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

A. F. Labossiere, Sobalian Coefficients.

A. F. Labossiere, Miscellaneous.

T. Mansour and J. West, Avoiding 2-letter signed patterns.

Hisanori Mishima, Factorizations of many number sequences

Hisanori Mishima, Factorizations of many number sequences

Jon Perry, Sum of Factorials

Alexsandar Petojevic, The Function vM_m(s; a; z) and Some Well-Known Sequences, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7

Eric Weisstein's World of Mathematics, Left Factorial

Index entries for sequences related to factorial numbers

a(n) = n*a(n-1)-(n-1)*a(n-2) - Henry Bottomley (se16(AT)btinternet.com), Feb 28 2001

Sequence is given by 1+1[1+2[1+3[1+4[1+..., terminating in n[1]..]. - Jon Perry (perry(AT)globalnet.co.uk), Jun 01 2004

a(n) = Sum[P(n, k) / C(n, k) {k=0...n-1}] - Ross La Haye (rlahaye(AT)new.rr.com), Sep 20 2004

!n = n + C(n-2, 1) + 3*C(n-3, 1) + C(n-2, 2) + 9*C(n-4, 1) + 8*C(n-3, 2) + 33*C(n-5, 1) + 46*C(n-4, 2) + 8*C(n-3, 3) + 153*C(n-6, 1) + 272*C(n-5, 2) + 101*C(n-4, 3) + 3*C(n-3, 4) + 873*C(n-7, 1) + 1796*C(n-6, 2) + 975*C(n-5, 3) + 114*C(n-4, 4) + 5913*C(n-8, 1) + 13424*C(n-7, 2) + 9175*C(n-6, 3) + 1935*C(n-5, 4) + 65*C(n-4, 5) + 46233*C(n-9, 1) + ..... . - Andre F. Labossiere (boronali(AT)laposte.net), Feb 03 2005

E.g.f.: (Ei(1)-Ei(1-x))*exp(-1+x) where Ei(x) is the exponential integral - Djurdje Cvijovic and Aleksandar Petojevic (apetoje(AT)ptt.yu), Apr 11 2000

a(n) = Integral_{x=0..infinity} [(x^n-1)/(x-1)]*exp(-x) dx - Gerald McGarvey (gerald.mcgarvey(AT)comcast.net), Oct 12 2007

A007489(n)=!(n+1)+1=a(n+1)+1 - Artur Jasinski, Nov 08 2007

Starting (1, 2, 4, 10, 34, 154,...), = row sums of triangle A135722 - Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 25 2007

a(n) = a(n-1) + (n-1)! for n >= 2. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Jun 16 2009]

!5 = 0!+1!+2!+3!+4! = 1+1+2+6+24 = 34.

A003422 := proc(n) local k; add(k!, k=0..n-1); end;

Table[Sum[i!, {i, 0, n - 1}], {n, 0, 20}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Mar 31 2006

Equals A007489 - 1. Cf. A000142, A014144, A005165.

Twice A014288. See also A049782, A100612.

Cf. A102639, A102411, A102412, A101752, A094216, A094638, A008276, A000166, A000110, A000204, A000045, A000108.

Cf. A135722.

Sequence in context: A154219 A089476 A006397 this_sequence A117402 A109455 A156800

Adjacent sequences: A003419 A003420 A003421 this_sequence A003423 A003424 A003425

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com), R. K. Guy