A088327 G.f.: exp(Sum_{k>=1} B(x^k)/k), where B(x) = x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + ... = (C(x)-1)/x and C is the g.f. for the Catalan numbers A000108.

1, 1, 3, 8, 25, 77, 256, 854, 2940, 10229, 36124, 128745, 463137, 1677816, 6118165, 22432778, 82660369, 305916561, 1136621136, 4238006039, 15852603939, 59471304434, 223704813807, 843547443903, 3188064830876, 12074092672950, 45816941923597, 174173975322767

Offset: 0

Author

N. J. A. Sloane, Nov 06 2003

Keywords

• nonn

Comments

a(n) is the number of forests of rooted plane binary trees (each node has outdegree = 0 or 2) where the trees have a total of n internal nodes. Cf. A222006. - Geoffrey Critzer, Feb 26 2013

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Row sums of A275431.

Cf. A222006, A304787.

Programs

Magma

m:=35;
f:= func< x | (&*[1/(1-x^j)^Catalan(j): j in [1..m+2]]) >;
R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( f(x) )); // G. C. Greubel, Dec 12 2022

Maple

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
       binomial(2*d, d)/(d+1), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Sep 10 2012

Mathematica

With[{nn=35}, CoefficientList[Series[Product[1/(1-x^i)^CatalanNumber[i], {i,nn}], {x,0,nn}], x]] (* Geoffrey Critzer, Feb 26 2013 *)

SageMath

# uses[EulerTransform from A166861]
b = EulerTransform(lambda n: binomial(2*n, n)/(n+1))
print([b(n) for n in range(30)]) # Peter Luschny, Nov 11 2020

Formula

Euler transform of Catalan numbers (A000108). - Franklin T. Adams-Watters, Mar 01 2006

a(n) ~ c * 4^n / n^(3/2), where c = exp(Sum_{k>=1} (-2 + 4^k - 4^k*sqrt(1 - 4^(1-k)))/(2*k) ) / sqrt(Pi) = 1.60022306097485382475864802335610662545... - Vaclav Kotesovec, Mar 21 2021