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A052854 Number of forests of ordered trees on n total nodes. 4
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, 119725350703472 (list; graph; refs; listen; history; text; 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 multisets of B with a total of n points.

LINKS

T. D. Noe and Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 201 terms from T. D. Noe)

Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 40.

P. Flajolet et al., A hybrid of Darboux's method and singularity analysis in combinatorial asymptotics, arXiv:math/0606370 [math.CO], 2006.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 822

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, 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 ~ 1.71603053492228196404746... (see A246949).

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

# third Maple program:

with(numtheory):

b:= proc(n) option remember; binomial(2*n, n) end:

a:= proc(n) option remember; `if`(n=0, 1, add(add(

      b(d-1), d=divisors(j))*a(n-j), j=1..n)/n)

    end:

seq(a(n), n=0..35);  # Alois P. Heinz, Mar 10 2015

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] (* Jean-Fran├žois 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.

Cf. A246949.

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

EXTENSIONS

Better title from Geoffrey Critzer, Feb 22 2013

Minor edits, Vaclav Kotesovec, May 13 2014

STATUS

approved

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Last modified August 18 22:15 EDT 2017. Contains 290768 sequences.