A209319 Number of functions f:{1,2,...,n}->{1,2,...,n} whose cycle lengths are <= 2.

1, 1, 4, 25, 218, 2451, 33832, 554527, 10535100, 227790505, 5525843696, 148673435769, 4394818486456, 141611317636075, 4940870266568160, 185595910032346111, 7468517348971708688, 320562141349559055633, 14619577651630443611200, 706025600924216704982425

Offset: 0

Author

Geoffrey Critzer, Jan 19 2013

Keywords

• nonn

Examples

			a(3) = 25 because there are 27 functions from {1,2,3} into itself but 2 of these have cycle length of 3: 2,3,1, and 3,1,2.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..350

Crossrefs

Column k=2 of A246522.

Programs

Maple

T:= -LambertW(-x):
egf:= exp(T + T^2/2):
a:= n-> n!*coeff(series(egf, x, n+1), x, n):
seq(a(n), n=0..20);  # Alois P. Heinz, Jan 19 2013

Mathematica

nn=20;t=Sum[n^(n-1)x^n/n!,{n,1,nn}];Range[0,nn]!CoefficientList[Series[Exp[Sum[t^i/i,{i,1,2}]],{x,0,nn}],x]

Formula

E.g.f.: exp(T(x) + T(x)^2/2) = A(T(x)) where A(x) is the e.g.f. for A000085 and T(x) is the e.g.f. for A000169.

a(n) ~ 2*exp(3/2)*n^(n-1). - Vaclav Kotesovec, Sep 30 2013