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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.

%I A065513 #21 Sep 08 2022 08:45:04
%S A065513 2,24,300,4320,72030,1376256,29760696,720000000,19292299290,
%T A065513 567575838720,18197320924068,631732166467584,23613833496093750,
%U A065513 945755921747804160,40410678374256222960,1835086247681868693504,88263072551692077310386,4482662400000000000000000
%N 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.
%H A065513 Alois P. Heinz, <a href="/A065513/b065513.txt">Table of n, a(n) for n = 3..150</a>
%F A065513 E.g.f.: T^3/3 where T=T(x) is Euler's tree function (see A000169).
%F A065513 a(n) = (n-1)*(n-2)*n^(n-3). - _Vaclav Kotesovec_, Oct 05 2013
%F A065513 a(n) = 2*A053507(n). - _Vaclav Kotesovec_, Oct 07 2016
%e A065513 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)
%p A065513 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);
%t A065513 nn = 18; t = Sum[n^(n - 1) x^n/n!, {n, 1, nn}];
%t A065513 Range[0, nn]! CoefficientList[Series[2 t^3/3!, {x, 0, nn}], x] (* _Geoffrey Critzer_, Aug 14 2013 *)
%o A065513 (PARI) for(n=3,50, print1((n-1)*(n-2)*n^(n-3), ", ")) \\ _G. C. Greubel_, Nov 14 2017
%o A065513 (Magma) [(n-1)*(n-2)*n^(n-3): n in [3..50]]; // _G. C. Greubel_, Nov 14 2017
%Y A065513 Cf. A000169 (unique cycle is length 1), A053506 (unique cycle has length 2).
%Y A065513 Column k=3 of A201685.
%K A065513 nonn
%O A065513 3,1
%A A065513 _Len Smiley_, Nov 27 2001