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
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 A published 2004 proof was retracted in 2011.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 byJeppe 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
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. -Aleksandar Petojevic,Jun 29 2018
This sequence converges in the p-adic topology, for every prime number p. -Harry Richman,Aug 13 2024
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.
Aleksandar Petojević, Marjana Gorjanac Ranitović, Dragan Rastovac, and Milinko Mandić,The Golden Ratio, Factorials, and the Lambert W Function,Journal of Integer Sequences, Vol. 27 (2024), Article 24.5.7.
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.
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 andAleksandar Petojevic,Apr 11 2000
a(n) = Integral_{x = 0..oo} [(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 byAntti Karttunen,Nov 29 2013
Starting (1, 2, 4, 10, 34, 154,...), = row sums of triangleA135722.-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
FromSergei 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. ofA153229.
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. ofA136580.(End)
a(n) = (-1)^(n+1)*C(n-1, -1) where C(n, x) are the Charlier polynomials (with parameter a=1) as given inA137338.(Evaluation at x = 1 givesA232845.) -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 +...
FromArkadiusz 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 polynomialsA137338:
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
defA003422_gen(): # generator of terms
yield from (0, 1)
c, f = 1, 1
for n in count(1):
yield (c:= c + (f:= f*n))
(Python)
def a(n):
if n == 0: return 0
s = f = 1
for k in range(1, n):
f *= k
s += f
return round(s)
print([a(n) for n in range(24)]) #Peter Luschny,Mar 05 2024
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
STATUS
approved