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A003422
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Left factorials: !n = Sum k!, k=0..n-1.
(Formerly M1237)
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63
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0, 1, 2, 4, 10, 34, 154, 874, 5914, 46234, 409114, 4037914, 43954714, 522956314, 6749977114, 93928268314, 1401602636314, 22324392524314, 378011820620314, 6780385526348314, 128425485935180314, 2561327494111820314, 53652269665821260314
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| 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
a(1)=0, a(n+1) is partial sum of n-th factorial number. [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Oct 16 2010]
Also, numbers left over after the following sieving process: At step 1, keep all numbers of the set N={0,1,2,...}. In step 2, keep only every second number after a(2)=2: N'={0,1,2,4,6,8,10,...}. In step 3, keep every third of the numbers following a(3)=4, N"={0,1,2,4,10,16,22,...}. In step 4, keep every fourth of the numbers beyond a(4)=10: {0,1,2,4,10,34,58,...}, and so on. [From M. F. Hasler (MHasler(AT)univ-ag.fr), Oct 28 2010]
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REFERENCES
| R. K. Guy, Unsolved Problems Number Theory, Section B44.
D. Kurepa, On the left factorial function !n. Math. Balkanica 1 1971 147-153.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| 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.
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
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FORMULA
| 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
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]
E.g.f. A(x) satisfies the differential equation A'(x)=A(x)+1/(1-x) [From Vladimir Kruchinin (kru(AT)ie.tusur.ru), Jan 19 2011]
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EXAMPLE
| !5 = 0!+1!+2!+3!+4! = 1+1+2+6+24 = 34.
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MAPLE
| A003422 := proc(n) local k; add(k!, k=0..n-1); end;
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MATHEMATICA
| Table[Sum[i!, {i, 0, n - 1}], {n, 0, 20}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Mar 31 2006
Join[{0}, Accumulate[Range[0, 25]!]] (* From Harvey P. Dale, Nov 19 2011 *)
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PROG
| (PARI) a(n)=sum(k=0, n-1, k!) \\ Charles R Greathouse IV, Jun 15 2011
(Haskell)
a003422 n = a003422_list !! n
a003422_list = scanl (+) 0 a000142_list
-- Reinhard Zumkeller, Dec 27 2011
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CROSSREFS
| 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, A135722.
Sequence in context: A173489 A089476 A006397 * A117402 A109455 A189591
Adjacent sequences: A003419 A003420 A003421 * A003423 A003424 A003425
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), R. K. Guy
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