OFFSET
0,4
COMMENTS
Conjecture: for n > 2, smallest prime divisor of a(n) > n. - Gerald McGarvey, Jun 19 2004
Rebuttal: This is not true; see Zivkovic link (Math. Comp. 68 (1999), pp. 403-409) has demonstrated that 3612703 divides a(n) for all n >= 3612702. - Paul Jobling, Oct 18 2004
Conjecture: For n>1, a(n) is the number of lattice paths from (0,0) to (n+1,0) that do not cross above y=x or below the x-axis using up-steps +(1,a) and down-steps +(1,-b) where a and b are positive integers. For example, a(3) = 5: [(1,1)(1,1)(1,1)(1,-3)], [(1,1)(1,-1)(1,3)(1,-3)], [(1,1)(1,-1)(1,2)(1,-2)], [(1,1)(1,-1)(1,1)(1,-1)] and [(1,1)(1,1)(1,-1)(1,-1)]. - Nicholas Ham, Aug 23 2015
Ham's claim is true for n=2. We proceed with a proof for n>2 by induction. On the j-th step, from (j-1,y) to (j,y'), there are j options for y': 0, 1, ..., y-1, y+1, ..., j. Thus there are n! possible paths from (0,0) to x=n that stay between y=0 and y=x. (Then the final step is determined.) However, because +(1,0) is not an allowable step, we cannot land on (n,0) on the n-th step. Therefore, the number of acceptable lattice paths is n! - a(n-1). - Danny Rorabaugh, Nov 30 2015
REFERENCES
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B10, pp. 152-153.
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
Richard K. Guy, Letter to N. J. A. Sloane, Sep 25 1986.
Richard K. Guy, Letter to N. J. A. Sloane, 1987
Richard K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712.
Richard K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
Hisanori Mishima, Factorizations of many number sequences: 103 and 130.
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 Wegrzynowski, Séries de factorielles.
Eric Weisstein's World of Mathematics, Alternating Factorial and Factorial.
Miodrag Živković, The number of primes Sum_{i=1..n} (-1)^(n-i)*i! is finite, Math. Comp. 68 (1999), pp. 403-409.
FORMULA
a(0) = 0, a(n) = n! - a(n-1) for n > 0; also a(n) = n*a(n-2) + (n-1)*a(n-1) for n > 1. Sum_{n>=1} Pi^n/a(n) ~ 30.00005. - Gerald McGarvey, Jun 19 2004
E.g.f.: 1/(1-x) + exp(-x)*(e*(Ei(1,1)-Ei(1,1-x)) - 1). - Robert Israel, Dec 01 2015
a(n) = (-1)^n*(exp(1)*(gamma(n+2)*gamma(-1-n,1)*(-1)^n +Ei(1))-1). - Gerry Martens, May 22 2018
Sum_{n>=1} 1/a(n) = A343187. - Amiram Eldar, Jun 01 2023
MAPLE
A005165 := proc(n) local i; add((-1)^(n-i)*i!, i=1..n); end;
MATHEMATICA
nn=25; With[{fctrls=Range[nn]!}, Table[Abs[Total[Times@@@Partition[ Riffle[ Take[ fctrls, n], {1, -1}], 2]]], {n, nn}]] (* Harvey P. Dale, Dec 10 2011 *)
a[0] = 0; a[n_] := n! - a[n - 1]; Array[a, 26, 0] (* Robert G. Wilson v, Aug 06 2012 *)
RecurrenceTable[{a[n] == n! - a[n - 1], a[0] == 0}, a, {n, 0, 20}] (* Eric W. Weisstein, Jul 27 2017 *)
AlternatingFactorial[Range[0, 20]] (* Eric W. Weisstein, Jul 27 2017 *)
a[n_] = (-1)^n (Exp[1]((-1)^n Gamma[-1-n, 1] Gamma[2+n] - ExpIntegralEi[-1]) - 1)
Table[a[n] // FullSimplify, {n, 0, 20}] (* Gerry Martens, May 22 2018 *)
PROG
(PARI) a(n)=if(n<0, 0, sum(k=0, n-1, (-1)^k*(n-k)!))
(Python)
a = 0
f = 1
for n in range(1, 33):
print(a, end=", ")
f *= n
a = f - a
# Alex Ratushnyak, Aug 05 2012
(PARI) first(m)=vector(m, j, sum(i=0, j-1, ((-1)^i)*(j-i)!)) \\ Anders Hellström, Aug 23 2015
(PARI) a(n)=round((-1)^n*(exp(1)*(gamma(n+2)*incgam(-1-n, 1)*(-1)^n +eint1(1))-1)) \\ Gerry Martens, May 22 2018
(Haskell)
a005165 n = a005165_list !! n
a005165_list = 0 : zipWith (-) (tail a000142_list) a005165_list
-- Reinhard Zumkeller, Jul 21 2013
(GAP) List([0..30], n->Sum([1..n], i->(-1)^(n-i)*Factorial(i))); # Muniru A Asiru, Jun 01 2018
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
STATUS
approved