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A005165 Alternating factorials: n! - (n-1)! + (n-2)! - ... 1!.
(Formerly M3892)
13
0, 1, 1, 5, 19, 101, 619, 4421, 35899, 326981, 3301819, 36614981, 442386619, 5784634181, 81393657019, 1226280710981, 19696509177019, 335990918918981, 6066382786809019, 115578717622022981, 2317323290554617019, 48773618881154822981 (list; graph; refs; listen; history; text; internal format)
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

R. K. Guy, Unsolved Problems in Number Theory, B43.

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

R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712.

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

E. Wegrzynowski, Series de factorielles

Eric Weisstein's World of Mathematics, Alternating Factorial and Factorial

M. 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(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

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 *)

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,

    f *= n

    a = f - a

# from 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

(Haskell)

a005165 n = a005165_list !! n

a005165_list = 0 : zipWith (-) (tail a000142_list) a005165_list

-- Reinhard Zumkeller, Jul 21 2013

CROSSREFS

Cf. A000142, A001272, A003422, A071828.

Sequence in context: A106958 A146144 A162292 * A071828 A158615 A088180

Adjacent sequences:  A005162 A005163 A005164 * A005166 A005167 A005168

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified May 29 03:15 EDT 2016. Contains 273487 sequences.