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 A005165 Alternating factorials: n! - (n-1)! + (n-2)! - ... 1!. (Formerly M3892) 58
 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, Letter to N. J. A. Sloane, Sep 25 1986. R. K. Guy, Letter to N. J. A. Sloane, 1987 R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. R. 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 E. Wegrzynowski, Séries 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. 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 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,     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 Cf. A000142, A001272, A003422, A071828, A303697. Sequence in context: A106958 A146144 A162292 * A071828 A280067 A158615 Adjacent sequences:  A005162 A005163 A005164 * A005166 A005167 A005168 KEYWORD nonn,easy,nice AUTHOR STATUS approved

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Last modified November 14 11:04 EST 2018. Contains 317182 sequences. (Running on oeis4.)