A088436 -id:A088436 - cp's OEIS frontend

A114320 Triangle T(n,k) = number of permutations of n elements with k 2-cycles.

1, 1, 1, 1, 3, 3, 15, 6, 3, 75, 30, 15, 435, 225, 45, 15, 3045, 1575, 315, 105, 24465, 12180, 3150, 420, 105, 220185, 109620, 28350, 3780, 945, 2200905, 1100925, 274050, 47250, 4725, 945, 24209955, 12110175, 3014550, 519750, 51975, 10395, 290529855

Offset: 0

Vladeta Jovovic, Feb 05 2006

Row n has 1+floor(n/2) terms. Row sums yield the factorials (A000142). Sum(k*T(n,k),k>0)=n!/2 for n>=2. - Emeric Deutsch, Feb 17 2006

			T(3,1) = 3 because we have (1)(23), (12)(3) and (13)(2).
Triangle begins:
    1;
    1;
    1,   1;
    3,   3;
   15,   6,   3;
   75,  30,  15;
  435, 225,  45,  15;
  ...

Alois P. Heinz, Rows n = 0..200, flattened

Cf. A008290, A000266, A000090, A088436, A000138, A060725, A060726, A086659, A105422, A105114.

G:= exp((y-1)*x^2/2)/(1-x): Gser:= simplify(series(G,x=0,15)): P[0]:=1: for n from 1 to 12 do P[n]:= n!*coeff(Gser,x^n) od: for n from 0 to 12 do seq(coeff(y*P[n], y^j), j=1..1+floor(n/2)) od;  # yields sequence in triangular form - Emeric Deutsch, Feb 17 2006

d = Exp[-x^2/2!]/(1 - x);f[list_] := Select[list, # > 0 &]; Flatten[Map[f, Transpose[Table[Range[0, 10]!CoefficientList[Series[x^(2 k)/(2^k k!) d, {x, 0, 10}], x], {k, 0, 5}]]]]  (* Geoffrey Critzer, Nov 29 2011 *)

E.g.f.: exp((y-1)*x^2/2)/(1-x). More generally, e.g.f. for number of permutations of n elements with k m-cycles is exp((y-1)*x^m/m)/(1-x).

T(n,k) = n!/(2^k*k!) * Sum_{j=0..floor(n/2)-k} (-1/2)^j/j!. - Alois P. Heinz, Nov 30 2011

More terms from Emeric Deutsch, Feb 17 2006

A027616 Number of permutations of n elements containing a 2-cycle.

Original entry on oeis.org

0, 0, 1, 3, 9, 45, 285, 1995, 15855, 142695, 1427895, 15706845, 188471745, 2450132685, 34301992725, 514529890875, 8232476226975, 139952095858575, 2519137759913775, 47863617438361725, 957272348112505425, 20102719310362613925, 442259824841726816925, 10171975971359716789275

Offset: 0

Joe Keane (jgk(AT)jgk.org)

nonn

Robert Israel, Table of n, a(n) for n = 0..449
Larry Carter and Stan Wagon, The Mensa Correctional Institute, The American Mathematical Monthly 125.4 (2018): 306-319.

Cf. A000266, A088436, A114320.

Column k=2 of A293211.

A027616:= func< n | Factorial(n)*(1- (&+[(-1/2)^j/Factorial(j): j in [0..Floor(n/2)]]) ) >;
[A027616(n): n in [0..30]]; // G. C. Greubel, Aug 05 2022

S:= series((1-exp(-x^2/2))/(1-x), x, 101):
seq(coeff(S,x,j)*j!,j=0..100); # Robert Israel, May 12 2016

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 *)

a(n) = n! * (1 - sum(k=0,floor(n/2), (-1)^k / (2^k * k!) ) );
/* Joerg Arndt, Oct 20 2012 */

N=33; x='x+O('x^N);
v=Vec( 'a0 + serlaplace( (1-exp(-x^2/2))/(1-x) ) );
v[1]-='a0;  v
/* Joerg Arndt, Oct 20 2012 */

def A027616(n): return factorial(n)*(1-sum((-1/2)^k/factorial(k) for k in (0..(n//2))))
[A027616(n) for n in (0..30)] # G. C. Greubel, Aug 05 2022

E.g.f.: (1 - exp(-x^2/2)) / (1-x).
a(n) = n! * ( 1 - Sum_{k=0..floor(n/2)} (-1)^k / (2^k * k!) ).
a(n) + A000266(n) = n!. - Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 09 2003
Limit_{n -> oo} a(n)/n! = 1 - e^(-1/2) = 1 - A092605. - Michel Marcus, Aug 08 2013

Added more terms, Geoffrey Critzer, Oct 20 2012

A178669 The number of permutations of [n] with 2 cycles of length 2.

Original entry on oeis.org

0, 3, 15, 45, 315, 3150, 28350, 274050, 3014550, 36330525, 472296825, 6609317715, 99139765725, 1586293008300, 26966981141100, 485404420000500, 9222683980009500, 184453709062998375

Offset: 3

Paul Weisenhorn, Jun 02 2010

nonn

			a(4)=3 counts the 3 permutations (2143), (3412), (4321) with 2 cycles
of length 2

Vincenzo Librandi, Table of n, a(n) for n = 3..200

Cf. A088436 (k=1 cycle), A000266 (k=0 cycle).

A178669 := proc(n) local k ; k :=2 ; n!*add( (-1)^j/(j-k)!/2^j/k!,j=k..n/2) ; end proc:
seq(A178669(n),n=3..20) ;

d=Exp[-x^2/2]/(1-x); Range[0,20]! CoefficientList[Series[(3x^4/4! )d, {x,0,20}], x] (* Geoffrey Critzer, Nov 29 2011 *)

a(n)=n!*sum_{j=k.. [n/2]} (-1)^j/((j-k)!*2^j*k!). E.g.f. = exp(-z^2/2)*z^(2*k) / ((1-z)*2^k*k!), where k is the number of cycles of length 2.

a(n) ~ n! * exp(-1/2)/8. - Vaclav Kotesovec, Mar 20 2014

Typo in a(18) corrected by Vincenzo Librandi, Mar 21 2014

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A114320 Triangle T(n,k) = number of permutations of n elements with k 2-cycles.

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A027616 Number of permutations of n elements containing a 2-cycle.

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A178669 The number of permutations of [n] with 2 cycles of length 2.

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