A162970 Number of 2-cycles in all involutions of {1,2,...,n}.

0, 1, 3, 12, 40, 150, 546, 2128, 8352, 34380, 144100, 626736, 2784288, 12753832, 59692920, 286857600, 1407536896, 7069630608, 36217682352, 189489626560, 1010037302400, 5488251406176, 30348031302688, 170812160339712

Offset

1,3

Links

Alois P. Heinz, Table of n, a(n) for n = 1..300

Formula

a(n) = (1/2)*n*(n-1)*I(n-2) for n>=2, where I(n)=A000085(n) is the number of involutions of {1,2,...,n}.

Rec. rel.: a(n) = [n/(n-2)][a(n-1) + (n-1)a(n-2)], a(1)=0, a(2)=1.

E.g.f.: x^2/2 * exp(x+x^2/2).

a(n) ~ sqrt(2)/4 * n^(n/2+1)*exp(sqrt(n)-n/2-1/4) * (1-17/(24*sqrt(n))). - Vaclav Kotesovec, Aug 15 2013

Example

a(3) = 3 because in (1)(2)(3), (1)(23), (12)(3), (13)(2) we have three 2-cycles.

Maple

a[1] := 0: a[2] := 1: for n from 3 to 27 do a[n] := n*(a[n-1]+(n-1)*a[n-2])/(n-2) end do: seq(a[n], n = 1 .. 27);

Mathematica

Range[0, 20]! CoefficientList[ Series[x^2/2  Exp[x+x^2/2], {x, 0, 20}], x] // Rest

Crossrefs

Cf. A000085, A013989.

Sequence in context: A102839 A050182 A222610 * A126725 A053043 A263102

Adjacent sequences: A162967 A162968 A162969 * A162971 A162972 A162973

Keyword

nonn

Author

Emeric Deutsch, Jul 22 2009

Status

approved