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

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

Offset

3,2

Links

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

Formula

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

Example

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

Maple

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) ;

Mathematica

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 *)

Crossrefs

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

Sequence in context: A005560 A100747 A100737 * A110464 A261505 A331505

Adjacent sequences: A178666 A178667 A178668 * A178670 A178671 A178672

Keyword

nonn

Author

Paul Weisenhorn, Jun 02 2010

Extensions

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

Status

approved