A065513 Number of endofunctions of [n] with a cycle a->b->c->a and for all x in [n], some iterate f^k(x)=a.

2, 24, 300, 4320, 72030, 1376256, 29760696, 720000000, 19292299290, 567575838720, 18197320924068, 631732166467584, 23613833496093750, 945755921747804160, 40410678374256222960, 1835086247681868693504, 88263072551692077310386, 4482662400000000000000000

Offset: 3

Author

Len Smiley, Nov 27 2001

Keywords

• nonn

Examples

			a(4)=24: 1->2->3->1<-4; 2->3->1->2<-4; 3->1->2->3<-4 1->3->2->1<-4; 3->2->1->3<-4; 2->1->3->2<-4 (repeat with 1,2, then 3 excluded from cycle)
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 3..150

Crossrefs

Cf. A000169 (unique cycle is length 1), A053506 (unique cycle has length 2).

Column k=3 of A201685.

Programs

Magma

[(n-1)*(n-2)*n^(n-3): n in [3..50]]; // G. C. Greubel, Nov 14 2017

Maple

T := x->-LambertW(-x); a := []; f := series((T(x))^3/3,x,24); for m from 1 to 24 do a := [op(a),op(2*m-1,f)*(m+2)! ] od; print(a);

Mathematica

nn = 18; t = Sum[n^(n - 1) x^n/n!, {n, 1, nn}];
Range[0, nn]! CoefficientList[Series[2 t^3/3!, {x, 0, nn}], x] (* Geoffrey Critzer, Aug 14 2013 *)

PARI

for(n=3,50, print1((n-1)*(n-2)*n^(n-3), ", ")) \\ G. C. Greubel, Nov 14 2017

Formula

E.g.f.: T^3/3 where T=T(x) is Euler's tree function (see A000169).

a(n) = (n-1)*(n-2)*n^(n-3). - Vaclav Kotesovec, Oct 05 2013

a(n) = 2*A053507(n). - Vaclav Kotesovec, Oct 07 2016