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A003422 Left factorials: !n = Sum k!, k=0..n-1.
(Formerly M1237)
75
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; text; internal format)
OFFSET

0,3

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, Nov 29 2007

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. - M. F. Hasler, Oct 28 2010

If s(n) is a second order recurrence defined as s(0) = x, s(1) = y, s(n) = n*(s(n - 1) - s(n - 2)), n > 1, then s(n) = n*y - n*a(n - 1)*x. - Gary Detlefs, May 27 2012

a(n) is the number of lists of {1, .., n} with (1st element) = (smallest element) and (k-th element) <> (k-th smallest element) for k > 1, where a list means an ordered subset. a(4) = 10 because we have the lists: [1], [2], [3], [4], [1, 3, 2], [1, 4, 2], [1, 4, 3], [2, 4, 3], [1, 3, 4, 2], [1, 4, 2, 3]. Cf. A000262. - Geoffrey Critzer, Oct 04 2012

Consider a tree graph with 1 vertex. Add an edge to it with another vertex. Now add 2 edges with vertices to this vertex, and then 3 edges to each open vertex of the tree (not the first one!), and the next stage is to add 4 edges, and so on. The total number of vertices at each stage give this sequence (see example). - Jon Perry, Jan 27 2013

Additive version of the superfactorials A000178. - Jon Perry, Feb 09 2013

Repunits in the factorial number system (see links). - Jon Perry, Feb 17 2013

n|a(n) only for 1 and 2. - Robert G. Wilson v, Jun 15 2013

!n is not always squarefree for n > 3. Miodrag Zivkovic found that 54503^2 divides !26541. - Arkadiusz Wesolowski, Nov 20 2013

a(n) gives the position of A007489(n) in A227157. - Antti Karttunen, Nov 29 2013

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).

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.

Romeo Mestrovic, Variations of Kurepa's left factorial hypothesis, arXiv preprint arXiv:1312.7037, 2013

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, Left Factorial (Mathworld).

Eric Weisstein, Factorial sums (Mathworld).

Eric Weisstein, Repunit

Wikipedia, Factorial number system

Miodrag Zivkovic, The number of primes sum_{i=1..n} (-1)^(n-i)*i! is finite, Math. Comp. 68 (1999), pp. 403-409.

Index entries for sequences related to factorial numbers

FORMULA

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

Sequence is given by 1 + 1[1 + 2[1 + 3[1 + 4[1 + ..., terminating in n[1]..]. - Jon Perry, Jun 01 2004

a(n) = Sum[P(n, k) / C(n, k) {k = 0...n - 1}]. - Ross La Haye, 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, Oct 12 2007

A007489(n) = !(n + 1) - 1 = a(n + 1) - 1. - Artur Jasinski, Nov 08 2007. Typos corrected by Antti Karttunen, Nov 29 2013

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

a(n) = a(n - 1) + (n - 1)! for n >= 2. - Jaroslav Krizek, Jun 16 2009

E.g.f. A(x) satisfies the differential equation A'(x) = A(x) + 1/(1 - x). - Vladimir Kruchinin, Jan 19 2011

a(n + 1) = p(-1) where p(x) is the unique degree-n polynomial such that p(k) = A182386(k) for k = 0, 1, ..., n. - Michael Somos, Apr 27 2012

G.f.: x/(1-x)*Q(0), where Q(k)= 1 + (2*k + 1)*x/( 1 - 2*x*(k+1)/(2*x*(k+1) + 1/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 09 2013

G.f.: G(0)*x/(1-x)/2, where G(k)= 1 + 1/(1 - x*(k+1)/(x*(k+1) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 24 2013

G.f.: 2*x/(1-x)/G(0), where G(k)= 1 + 1/(1 - 1/(1 - 1/(2*x*(k+1)) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 29 2013

G.f.: W(0)*x/(1+sqrt(x))/(1-x), where W(k) = 1 + sqrt(x)/( 1 - sqrt(x)*(k+1)/(sqrt(x)*(k+1) + 1/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 17 2013

G.f.: B(x)*(1+x)/(1-x), where B(x) is the g.f. of A153229. - Sergei N. Gladkovskii, Aug 17 2013

G.f.: x/(1-x) + x^2/(1-x)/Q(0), where Q(k) = 1 - 2*x*(2*k+1) - x^2*(2*k+1)*(2*k+2)/( 1 - 2*x*(2*k+2) - x^2*(2*k+2)*(2*k+3)/Q(k+1) ) ; (continued fraction). - Sergei N. Gladkovskii, Sep 23 2013

G.f.: x*(1+x)*B(x), where B(x) is the g.f. of A136580. - Sergei N. Gladkovskii, Oct 22 2013

EXAMPLE

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

x + 2*x^2 + 4*x^3 + 10*x^4 + 34*x^5 + 154*x^6 + 874*x^7 + 5914*x^8 + 46234*x^9 + ...

Contribution from Arkadiusz Wesolowski, Aug 06 2012: (Start)

Illustration of initial terms:

.

. o        o         o            o                         o

.          o         o            o                         o

.                   o o          o o                       o o

.                              ooo ooo                   ooo ooo

.                                             oooo oooo oooo oooo oooo oooo

.

. 1        2         4            10                        34

.

(End)

The tree graph. The total number of vertices at each stage is 1, 2, 4, 10, ...

    0 0

    |/

    0-0

   /

0-0

   \

    0-0

    |\

    0 0

- Jon Perry, Jan 27 2013

MAPLE

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

MATHEMATICA

Table[Sum[i!, {i, 0, n - 1}], {n, 0, 20}] (* Stefan Steinerberger, Mar 31 2006 *)

Join[{0}, Accumulate[Range[0, 25]!]] (* Harvey P. Dale, Nov 19 2011 *)

a[0] = 0; a[1] = 1; a[n_] := a[n] = n*a[n - 1] - (n - 1)*a[n - 2]; Array[a, 23, 0] (* Robert G. Wilson v, Jun 15 2013 *)

a[n_] := (-1)^n*n!*Subfactorial[-n-1]-Subfactorial[-1]; Table[a[n] // FullSimplify, {n, 0, 22}] (* Jean-Fran├žois Alcover, Jan 09 2014 *)

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

CROSSREFS

Equals A007489(n-1)+1 for n>=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, A227157.

Cf. A000178.

Sequence in context: A089476 A220028 A006397 * A117402 A109455 A189591

Adjacent sequences:  A003419 A003420 A003421 * A003423 A003424 A003425

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane, R. K. Guy

STATUS

approved

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Last modified July 30 07:02 EDT 2014. Contains 245053 sequences.