0,3

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.

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

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

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.yu), 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.

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

(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))

Cf. A000108, A052805, A066768.

Sequence in context: A007580 A000085 A047653 this_sequence A096807 A003239 A116673

Adjacent sequences: A052851 A052852 A052853 this_sequence A052855 A052856 A052857

easy,nonn,nice

encyclopedia(AT)pommard.inria.fr, Jan 25 2000