A001540 Number of transpositions needed to generate permutations of length n.

0, 2, 8, 36, 184, 1110, 7776, 62216, 559952, 5599530, 61594840, 739138092, 9608795208, 134523132926, 2017846993904, 32285551902480, 548854382342176, 9879378882159186, 187708198761024552, 3754163975220491060, 78837443479630312280, 1734423756551866870182

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Examples

			a(5)=-1+5!(1+1/2!+1/4!)=-1+120+60+5=184.
		

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.

Crossrefs

Cf. A009179.

Programs

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

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

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 *)

PARI

a(n)=-1+n!*sum(k=0,floor(n/2),1/(2*k)!)

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

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) = |A009179(n)| - 1 = (n-1)*|A009179(n-1) - A009179(n-2)|.

a(n) ~ [cosh(1)*n!] - 1, where [x] is the floor of x. - Simon Plouffe, Nov 28 2018

Extensions

Edited by Ralf Stephan, Apr 16 2004