|
| |
|
|
A052854
|
|
Number of unordered forests on n nodes.
|
|
3
| |
|
|
1, 1, 2, 4, 10, 26, 77, 235, 758, 2504, 8483, 29203, 102030, 360442, 1285926, 4625102, 16754302, 61067430, 223803775, 824188993, 3048383517, 11318928477, 42176798315, 157664823501, 591109863049, 2222121888117, 8374151243258, 31630394287364
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,3
|
|
|
COMMENTS
| If B is a collection in which there are A000108(n-1) [Catalan numbers] things with n points, a(n) is the number of subsets of B with a total of n points.
|
|
|
REFERENCES
| Philippe Flajolet, Eric Fusy, Xavier Gourdon, Daniel Panario and Nicolas Pouyanne, A Hybrid of Darboux's Method and Singularity Analysis in Combinatorial Asymptotics, arXiv:math.CO/0606370
|
|
|
LINKS
| T. D. Noe, Table of n, a(n) for n=0..200
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 822
P. Flajolet et al., A hybrid of Darboux's method and singularity analysis in combinatorial asymptotics
|
|
|
FORMULA
| Euler transform of Catalan numbers C(n-1) (cf. A000108).
n*a(n)=Sum_{k=1..n} a(n-k)*b(k), b(k)=Sum_{d|k} binomial(2*d-2, d-1)=A066768(k). - Vladeta Jovovic (vladeta(AT)eunet.rs), Jan 17 2002
G.f.: 1/(Product_{k>0} (1-x^k)^C(k-1)) where C() is Catalan numbers.
G.f.: A(z) = prod_{n >= 1} (1-z^n)^(-A000108(n)) = exp(sum_{k >= 1} C(z^k)/k, where C(z) is the g.f. for the Catalan numbers.
a(n) ~ K 4^(n-1)/sqrt(pi n^3), where K ~ 1.71603.
|
|
|
MAPLE
| spec := [S, {B=Sequence(C), C=Prod(Z, B), S=Set(C)}, unlabeled]: seq(combstruct[count](spec, size=n), n=0..20); # version 1
spec := [ C, {B=Union(Z, Prod(B, B)), C=Set(B)}, unlabeled ]; [seq(combstruct[count](spec, size=n), n=0..40)]; # version 2
|
|
|
MATHEMATICA
| max = 27; f[x_] := 1/Product[ (1 - x^k)^CatalanNumber[k - 1], {k, 1, max}]; se = Series[f[x], {x, 0, max}]; CoefficientList[se, x] (* From Jean-François Alcover, Oct 05 2011, after g.f. *)
|
|
|
PROG
| (PARI) a(n)=if(n<0, 0, polcoeff(1/prod(k=1, n, (1-x^k+x*O(x^n))^((2*k-2)!/k!/(k-1)!)), n))
|
|
|
CROSSREFS
| Cf. A000108, A052805, A066768.
Sequence in context: A149817 A149818 A148101 * A148102 A179381 A096807
Adjacent sequences: A052851 A052852 A052853 * A052855 A052856 A052857
|
|
|
KEYWORD
| easy,nonn,nice
|
|
|
AUTHOR
| encyclopedia(AT)pommard.inria.fr, Jan 25 2000
|
| |
|
|
