A052505 Number of labeled 3-constrained functional graphs.

1, 3, 300, 141120, 182952000, 505008504000, 2547446533632000, 21222189199411200000, 271682221693022300160000, 5064076705822143609600000000, 131801391770668241689267200000000, 4632178742550388306775251353600000000

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

Table of n, a(n) for n=0..11.

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