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Number of permutations of n elements containing a 2-cycle.
4

%I #29 Aug 08 2022 08:29:54

%S 0,0,1,3,9,45,285,1995,15855,142695,1427895,15706845,188471745,

%T 2450132685,34301992725,514529890875,8232476226975,139952095858575,

%U 2519137759913775,47863617438361725,957272348112505425,20102719310362613925,442259824841726816925,10171975971359716789275

%N Number of permutations of n elements containing a 2-cycle.

%H Robert Israel, <a href="/A027616/b027616.txt">Table of n, a(n) for n = 0..449</a>

%H Larry Carter and Stan Wagon, <a href="https://www.jstor.org/stable/48663293">The Mensa Correctional Institute</a>, The American Mathematical Monthly 125.4 (2018): 306-319.

%F E.g.f.: (1 - exp(-x^2/2)) / (1-x).

%F a(n) = n! * ( 1 - Sum_{k=0..floor(n/2)} (-1)^k / (2^k * k!) ).

%F a(n) + A000266(n) = n!. - Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 09 2003

%F Limit_{n -> oo} a(n)/n! = 1 - e^(-1/2) = 1 - A092605. - _Michel Marcus_, Aug 08 2013

%p S:= series((1-exp(-x^2/2))/(1-x), x, 101):

%p seq(coeff(S,x,j)*j!,j=0..100); # _Robert Israel_, May 12 2016

%t nn=30; Table[n!,{n,0,nn}]-Range[0,nn]!CoefficientList[Series[Exp[-x^2/2]/(1-x),{x,0,nn}],x] (* _Geoffrey Critzer_, Oct 20 2012 *)

%o (PARI)

%o a(n) = n! * (1 - sum(k=0,floor(n/2), (-1)^k / (2^k * k!) ) );

%o /* _Joerg Arndt_, Oct 20 2012 */

%o (PARI)

%o N=33; x='x+O('x^N);

%o v=Vec( 'a0 + serlaplace( (1-exp(-x^2/2))/(1-x) ) );

%o v[1]-='a0; v

%o /* _Joerg Arndt_, Oct 20 2012 */

%o (Magma)

%o A027616:= func< n | Factorial(n)*(1- (&+[(-1/2)^j/Factorial(j): j in [0..Floor(n/2)]]) ) >;

%o [A027616(n): n in [0..30]]; // _G. C. Greubel_, Aug 05 2022

%o (SageMath)

%o def A027616(n): return factorial(n)*(1-sum((-1/2)^k/factorial(k) for k in (0..(n//2))))

%o [A027616(n) for n in (0..30)] # _G. C. Greubel_, Aug 05 2022

%Y Cf. A000266, A088436, A114320.

%Y Column k=2 of A293211.

%K nonn

%O 0,4

%A Joe Keane (jgk(AT)jgk.org)

%E Added more terms, _Geoffrey Critzer_, Oct 20 2012