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A003422 Left factorials: !n = Sum_{k=0..n-1} k!.
(Formerly M1237)
111
0, 1, 2, 4, 10, 34, 154, 874, 5914, 46234, 409114, 4037914, 43954714, 522956314, 6749977114, 93928268314, 1401602636314, 22324392524314, 378011820620314, 6780385526348314, 128425485935180314, 2561327494111820314, 53652269665821260314, 1177652997443428940314 (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) is the 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

Whether n|a(n) only for 1 and 2 remains an open problem. A published 2004 proof was retracted in 2011. This is sometimes known as Kurepa's conjecture. - Robert G. Wilson v, Jun 15 2013, corrected by Jeppe Stig Nielsen, Nov 07 2015.

!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

Matches the total domination number of the Bruhat graph from n = 2 to at least n = 5. - Eric W. Weisstein, Jan 11 2019

For the connection with Kurepa trees, see A. Petojevic, The {K_i(z)}_{i=1..oo} functions, Rocky Mtn. J. Math., 36 (2006), 1637-1650. - A. Petojevic, Jun 29 2018

REFERENCES

Richard 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), Article 00.1.5.

Bernd C. Kellner, Some remarks on Kurepa's left factorial, arXiv:math/0410477 [math.NT], 2004.

D. Kurepa, On the left factorial function !N, Math. Balkanica 1 (1971), 147-153. (Annotated scanned copy).

T. Mansour and J. West, Avoiding 2-letter signed patterns, arXiv:math/0207204 [math.CO], 2002.

Romeo Meštrović, Variations of Kurepa's left factorial hypothesis, arXiv preprint arXiv:1312.7037 [math.NT], 2013.

Romeo Meštrović, The Kurepa-Vandermonde matrices arising from Kurepa's left factorial hypothesis, Filomat 29:10 (2015), 2207-2215; DOI 10.2298/FIL1510207M.

Hisanori Mishima, Factorizations of many number sequences.

Hisanori Mishima, Factorizations of many number sequences

F. J. Papp, Letter to N. J. A. Sloane, Nov 1974.

Jon Perry, Sum of Factorials. [Broken link?]

Aleksandar Petojevic, On Kurepa's hypothesis for the left factorial, Filomat, Vol. 12, No. 1, 1998.

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.

Aleksandar Petojevic, The {K_i(z)}_{i=1..oo} functions, Rocky Mtn. J. Math., 36 (2006), 1637-1650.

Eric Weisstein's World of Mathematics, Factorial Sums.

Eric Weisstein's World of Mathematics, Left Factorial.

Eric Weisstein's World of Mathematics, 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

D-finite with recurrence: 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_{k=0..n-1} P(n, k) / C(n, k). - 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

From Sergei N. Gladkovskii, May 09 2013 to Oct 22 2013: (Start)

Continued fractions:

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))).

G.f.: G(0)*x/(1-x)/2 where G(k) = 1 + 1/(1 - x*(k+1)/(x*(k+1) + 1/G(k+1))).

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))).

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))).

G.f.: B(x)*(1+x)/(1-x) where B(x) is the g.f. of A153229.

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)).

G.f.: x*(1+x)*B(x) where B(x) is the g.f. of A136580. (End)

a(n) = (-1)^(n+1)*C(n-1, -1) where C(n, x) are the Charlier polynomials (with parameter a=1) as given in A137338. (Evaluation at x = 1 gives A232845.) - Peter Luschny, Nov 28 2018

a(n) = (a(n-3)*(n-2)^2*(n-3)! + a(n-1)^2)/a(n-2) (empirical). - Gary Detlefs, Feb 25 2022

a(n) = signum(n)/b(1,n) with b(i,n) = i - [i<n] * i/b(i+1,n). - Mohammed Bouras, Sep 07 2022

Sum_{n>=1} 1/a(n) = A357145. - Amiram Eldar, Oct 01 2022

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 + ...

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 proc:

# Alternative, using the Charlier polynomials A137338:

C := proc(n, x) option remember; if n > 0 then (x-n)*C(n-1, x) - n*C(n-2, x)

elif n = 0 then 1 else 0 fi end: A003422 := n -> (-1)^(n+1)*C(n-1, -1):

seq(A003422(n), n=0..22); # Peter Luschny, Nov 28 2018

# third Maple program:

a:= proc(n) option remember; `if`(n=0, 0, a(n-1)+(n-1)!) end:

seq(a(n), n=0..23); # Alois P. Heinz, Feb 24 2022

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-François Alcover, Jan 09 2014 *)

RecurrenceTable[{a[n] == n a[n - 1] - (n - 1) a[n - 2], a[0] == 0, a[1] == 1}, a, {n, 0, 10}] (* Eric W. Weisstein, Jan 11 2019 *)

Range[0, 20]! CoefficientList[Series[(ExpIntegralEi[1] - ExpIntegralEi[1 - x]) Exp[x - 1], {x, 0, 20}], x] (* Eric W. Weisstein, Jan 11 2019 *)

Table[(-1)^n n! Subfactorial[-n - 1] - Subfactorial[-1], {n, 0, 20}] // FullSimplify (* Eric W. Weisstein, Jan 11 2019 *)

Table[(I Pi + ExpIntegralEi[1] + (-1)^n n! Gamma[-n, -1])/E, {n, 0, 20}] // FullSimplify (* Eric W. Weisstein, Jan 11 2019 *)

PROG

(PARI) a003422(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

(Maxima) makelist(sum(k!, k, 0, n-1), n, 0, 20); /* Stefano Spezia, Jan 11 2019 */

(Python)

from itertools import count, islice

def A003422_gen(): # generator of terms

yield from (0, 1)

c, f = 1, 1

for n in count(1):

yield (c:= c + (f:= f*n))

A003422_list = list(islice(A003422_gen(), 20)) # Chai Wah Wu, Jun 22 2022

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, A000178, A137338, A232845, A357145.

Sequence in context: A006397 A297197 A297201 * A117402 A109455 A258948

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 November 26 20:27 EST 2022. Contains 358362 sequences. (Running on oeis4.)