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 A052505 Number of labeled 3-constrained functional graphs. 0
 1, 3, 300, 141120, 182952000, 505008504000, 2547446533632000, 21222189199411200000, 271682221693022300160000, 5064076705822143609600000000, 131801391770668241689267200000000, 4632178742550388306775251353600000000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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 LINKS INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 38 Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, page 131. FORMULA E.g.f.: -2/(-2+x*RootOf(-6*_Z+6*x+x*_Z^3)^2). 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] EXAMPLE 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. MAPLE 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); MATHEMATICA nn = 33; A[z]:= Sum[a[n] z^n, {n, 0, nn}]; sol = SolveAlways[0==Series[A[z] - z*(1+A[z]^3/6), {z, 0, nn}], z]; Select[Range[0, nn]!*Flatten[CoefficientList[Series[1/(1-zA[z]^2/2)/. sol, {z, 0, nn}], z]], # > 0 &] (* Geoffrey Critzer, Mar 14 2017 *) 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 *) CROSSREFS Cf. A036770. Sequence in context: A119049 A119067 A226500 * A307020 A071525 A157583 Adjacent sequences:  A052502 A052503 A052504 * A052506 A052507 A052508 KEYWORD easy,nonn AUTHOR encyclopedia(AT)pommard.inria.fr, Jan 25 2000 EXTENSIONS a(7)-a(11) from Geoffrey Critzer, Mar 14 2017 STATUS approved

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Last modified January 29 02:48 EST 2022. Contains 350672 sequences. (Running on oeis4.)