Search: id:A003422
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%I A003422 M1237
%S A003422 0,1,2,4,10,34,154,874,5914,46234,409114,4037914,43954714,522956314,
%T A003422 6749977114,93928268314,1401602636314,22324392524314,378011820620314,
%U A003422 6780385526348314,128425485935180314,2561327494111820314,53652269665821260314
%N A003422 Left factorials: !n = Sum k!, k=0..n-1.
%C A003422 Number of {12,12*,1*2,21*}- and {12,12*,21,21*}-avoiding signed permutations 
               in the hyperoctahedral group.
%C A003422 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
%D A003422 R. K. Guy, Unsolved Problems Number Theory, Section B44.
%D A003422 D. Kurepa, On the left factorial function !n. Math. Balkanica 1 1971 
               147-153.
%D A003422 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A003422 T. D. Noe, <a href="b003422.txt">Table of n, a(n) for n = 0..100</a>
%H A003422 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               Sequences realized by oligomorphic permutation groups</a>, J. Integ. 
               Seqs. Vol. 3 (2000), #00.1.5.
%H A003422 A. F. Labossiere, <a href="http://members.lycos.co.uk/sobalian/index.html">
               Sobalian Coefficients</a>.
%H A003422 A. F. Labossiere, <a href="http://members.lycos.co.uk/stereotomography/
               index.html">Miscellaneous</a>.
%H A003422 T. Mansour and J. West, <a href="http://arXiv.org/abs/math.CO/0207204">
               Avoiding 2-letter signed patterns</a>.
%H A003422 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/
               matha1/matha103.htm">Factorizations of many number sequences</a>
%H A003422 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/
               matha1/matha131.htm">Factorizations of many number sequences</a>
%H A003422 Jon Perry, <a href="http://www.users.globalnet.co.uk/~perry/maths/sumoffactorials/
               sumoffactorials.htm">Sum of Factorials</a>
%H A003422 Alexsandar Petojevic, <a href="http://www.cs.uwaterloo.ca/journals/JIS/
               index.html">The Function vM_m(s; a; z) and Some Well-Known Sequences</
               a>, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7
%H A003422 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               LeftFactorial.html">Left Factorial</a>
%H A003422 <a href="Sindx_Fa.html#factorial">Index entries for sequences related 
               to factorial numbers</a>
%F A003422 a(n) = n*a(n-1)-(n-1)*a(n-2) - Henry Bottomley (se16(AT)btinternet.com), 
               Feb 28 2001
%F A003422 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
%F A003422 a(n) = Sum[P(n, k) / C(n, k) {k=0...n-1}] - Ross La Haye (rlahaye(AT)new.rr.com), 
               Sep 20 2004
%F A003422 !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
%F A003422 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
%F A003422 a(n) = Integral_{x=0..infinity} [(x^n-1)/(x-1)]*exp(-x) dx - Gerald McGarvey 
               (gerald.mcgarvey(AT)comcast.net), Oct 12 2007
%F A003422 A007489(n)=!(n+1)+1=a(n+1)+1 - Artur Jasinski, Nov 08 2007
%F A003422 Starting (1, 2, 4, 10, 34, 154,...), = row sums of triangle A135722 - 
               Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 25 2007
%F A003422 a(n) = a(n-1) + (n-1)! for n >= 2. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), 
               Jun 16 2009]
%e A003422 !5 = 0!+1!+2!+3!+4! = 1+1+2+6+24 = 34.
%p A003422 A003422 := proc(n) local k; add(k!,k=0..n-1); end;
%t A003422 Table[Sum[i!, {i, 0, n - 1}], {n, 0, 20}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), 
               Mar 31 2006
%Y A003422 Equals A007489 - 1. Cf. A000142, A014144, A005165.
%Y A003422 Twice A014288. See also A049782, A100612.
%Y A003422 Cf. A102639, A102411, A102412, A101752, A094216, A094638, A008276, A000166, 
               A000110, A000204, A000045, A000108.
%Y A003422 Cf. A135722.
%Y A003422 Sequence in context: A154219 A089476 A006397 this_sequence A117402 A109455 
               A156800
%Y A003422 Adjacent sequences: A003419 A003420 A003421 this_sequence A003423 A003424 
               A003425
%K A003422 nonn,easy,nice
%O A003422 0,3
%A A003422 N. J. A. Sloane (njas(AT)research.att.com), R. K. Guy

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