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a(n) = number of endofunctions on [n] with a 4-cycle a->b->c->d->a and for any x in [n], some iterate f^k(x) = a.
3

%I #13 Dec 04 2015 11:33:04

%S 6,120,2160,41160,860160,19840464,504000000,14030763120,425681879040,

%T 13997939172360,496360987938816,18891066796875000,768426686420090880,

%U 33279382190563948320,1529238539734890577920,74326797938267012471904

%N a(n) = number of endofunctions on [n] with a 4-cycle a->b->c->d->a and for any x in [n], some iterate f^k(x) = a.

%H Alois P. Heinz, <a href="/A065888/b065888.txt">Table of n, a(n) for n = 4..150</a>

%F E.g.f.: T^4/4 where T = T(x) is Euler's tree function (see A000169).

%F a(n) = (n-1)*(n-2)*(n-3)*n^(n-4). - _Vaclav Kotesovec_, Oct 05 2013

%e a(4) = 6 : 3 [choices of 1's opposite in cycle] * 2 [choices of 1's image]

%t Rest[Rest[Rest[Rest[CoefficientList[Series[(LambertW[-x])^4/4, {x, 0, 20}], x]* Range[0, 20]!]]]] (* _Vaclav Kotesovec_, Oct 05 2013 *)

%t Table[(n-1)(n-2)(n-3)n^(n-4),{n,4,20}] (* _Harvey P. Dale_, Dec 04 2015 *)

%Y Cf. A000169 (1-cycle), A053506 (2-cycle), A065513 (3-cycle), A065889 (= A065888/2: underlying simple graphs).

%K nonn

%O 4,1

%A _Len Smiley_, Nov 27 2001