A005165 Alternating factorials: n! - (n-1)! + (n-2)! - ... 1!.

0, 1, 1, 5, 19, 101, 619, 4421, 35899, 326981, 3301819, 36614981, 442386619, 5784634181, 81393657019, 1226280710981, 19696509177019, 335990918918981, 6066382786809019, 115578717622022981, 2317323290554617019, 48773618881154822981

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

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.
• Index entries for sequences related to factorial numbers.

Crossrefs

Cf. A000142, A001272, A003422, A071828, A303697, A343187, A359808.

Programs

GAP

List([0..30],n->Sum([1..n],i->(-1)^(n-i)*Factorial(i))); # Muniru A Asiru, Jun 01 2018

Haskell

a005165 n = a005165_list !! n
a005165_list = 0 : zipWith (-) (tail a000142_list) a005165_list
-- Reinhard Zumkeller, Jul 21 2013

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 *)

PARI

a(n)=if(n<0,0,sum(k=0,n-1,(-1)^k*(n-k)!))

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

Python

a = 0
f = 1
for n in range(1, 33):
    print(a, end=",")
    f *= n
    a = f - a
# Alex Ratushnyak, Aug 05 2012

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