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%I #20 Dec 06 2019 15:55:50
%S 1,3,300,141120,182952000,505008504000,2547446533632000,
%T 21222189199411200000,271682221693022300160000,
%U 5064076705822143609600000000,131801391770668241689267200000000,4632178742550388306775251353600000000
%N Number of labeled 3-constrained functional graphs.
%C a(n) is the number of functions f:{1,2,...,3n}->{1,2,...,3n} such that the preimage of every element has cardinality 0 or 3. - _Geoffrey Critzer_, Mar 14 2017
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=38">Encyclopedia of Combinatorial Structures 38</a>
%H Philippe Flajolet and Robert Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/AnaCombi/anacombi.html">Analytic Combinatorics</a>, Cambridge Univ. Press, 2009, page 131.
%F E.g.f.: -2/(-2+x*RootOf(-6*_Z+6*x+x*_Z^3)^2).
%F Recurrence: {a(1)=0, (-9*n^4-54*n^3-117*n^2-108*n-36)*a(n)+(8*n+12)*a(n+3), a(2)=0, a(4)=0, a(3)=3, a(5)=0}. [interpolated with 0,0]
%e a(1) = 3 because there are 3 functions from {1,2,3} into {1,2,3} in which the preimage of every element in {1,2,3} is empty or contains 3 elements, namely the 3 constant functions.
%p spec := [S, {g=Union(Z, Prod(Z, Set(g, card=3))), S=Set(Cycle(Prod(Z, Set(g, card=2))))}, labeled]: seq(combstruct[count](spec, size=3*n), n=0..20);
%t nn = 33; A[z]:= Sum[a[n] z^n, {n, 0, nn}];
%t sol = SolveAlways[0==Series[A[z] - z*(1+A[z]^3/6), {z,0,nn}], z];
%t Select[Range[0, nn]!*Flatten[CoefficientList[Series[1/(1-zA[z]^2/2)/. sol, {z, 0, nn}], z]], # > 0 &] (* _Geoffrey Critzer_, Mar 14 2017 *)
%t Select[RecurrenceTable[{(-9*n^4-54*n^3-117*n^2-108*n-36)*a[n]+(8*n+12)*a[n+3]==0, a[0]==1, a[1]==0, a[2]==0}, a, {n,0,33}], # > 0 &] (* _Georg Fischer_, Dec 06 2019 *)
%Y Cf. A036770.
%K easy,nonn
%O 0,2
%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E a(7)-a(11) from _Geoffrey Critzer_, Mar 14 2017
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