OFFSET
0,3
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
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
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 *).
PROG
(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
(Magma)
m:=35;
f:= func< x | (&*[1/(1-x^j)^Catalan(j): j in [1..m+2]]) >;
R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!( f(x) )); // G. C. Greubel, Dec 12 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Nov 06 2003
STATUS
approved