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

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

!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.

Adjacent sequences: A003419 A003420 A003421 this_sequence A003423 A003424 A003425

Sequence in context: A056593 A089476 A006397 this_sequence A117402 A109455 A125859

nonn,easy,nice

njas, R. K. Guy