OFFSET
1,2
REFERENCES
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..450 (first 100 terms from T. D. Noe)
R. J. Ord-Smith, Generation of permutation sequences: Part 1, Computer J., 13 (1970), 151-155.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
FORMULA
E.g.f.: cosh(x)/(1-x) - exp(x).
Recurrence: a(n) = n*a(n-1) + n - (n mod 2).
a(n) = -1 + n!*Sum{k=0..floor(n/2)} 1/(2*k)! = -1 + round(n! * cosh(1)).
a(n) ~ [cosh(1)*n!] - 1, where [x] is the floor of x. - Simon Plouffe, Nov 28 2018
EXAMPLE
a(5)=-1+5!(1+1/2!+1/4!)=-1+120+60+5=184.
MAPLE
a := n -> (exp(1)*GAMMA(1 + n, 1) + exp(-1)*GAMMA(1 + n, -1))/2 - 1:
seq(simplify(a(n)), n=1..20); # Peter Luschny, Dec 05 2018
MATHEMATICA
With[{nn=20}, Rest[CoefficientList[Series[Cosh[x]/(1-x)-Exp[x], {x, 0, nn}], x]Range[0, nn]!]] (* Harvey P. Dale, Mar 04 2013 *)
PROG
(PARI) a(n)=-1+n!*sum(k=0, floor(n/2), 1/(2*k)!)
(J) a001540=:13 : '<:+/(!y)%!+:i.>:<.-:y' NB. Stephen Makdisi, May 02 2018
(Magma) [-1 + (&+[Factorial(n)/Factorial(2*k): k in [0..Floor(n/2)]]): n in [1..20]]; // G. C. Greubel, Nov 28 2018
(Sage) [-1 + factorial(n)*sum(1/factorial(2*k) for k in range(floor((n+2)/2))) for n in (1..20)] # G. C. Greubel, Nov 28 2018
(GAP) a:=[0];; for n in [2..20] do a[n]:=n*a[n-1]+n-(n mod 2); od; a; # Muniru A Asiru, Dec 05 2018
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
EXTENSIONS
Edited by Ralf Stephan, Apr 16 2004
STATUS
approved