login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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. 8
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 (list; graph; refs; listen; history; text; internal format)
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
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
Row sums of A275431.
Sequence in context: A022553 A292884 A148789 * A148790 A148791 A148792
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Nov 06 2003
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 19 03:33 EDT 2024. Contains 370952 sequences. (Running on oeis4.)