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!)
A286955 n-vertex sequences of plane forests with nondecreasing numbers of trees. 3
1, 1, 3, 9, 29, 96, 326, 1127, 3952, 14019, 50208, 181275, 659039, 2410433, 8862750, 32739168, 121443136, 452167865, 1689237104, 6330103627, 23787215202, 89616350271, 338417312294, 1280739676563, 4856711761475, 18451630811041, 70223495698892, 267691953822783 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Enumerates Part[Cat], the substitution of Cat for atoms of Part, where Part is the set of integer partitions (A000041), and Cat is any set counted by the 1-based Catalan numbers (A000108 shifted).

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

FORMULA

G.f.: Product_{k>0} 1/(1 - ((1 - sqrt(1 - 4*x))/2)^k), the composition of the g.f. for A000041 with x times the g.f. for A000108.

a(n) ~ c * 4^n / n^(3/2), where c = 2.680434829690402658212615372294526133126515771886321123341424399596963885434... - Vaclav Kotesovec, Jun 02 2018

EXAMPLE

a(3) = 9, consisting of (1,1,1), (1,2), (2,1), (3a), (3b), (1)(1,1), (1)(2), (2)(1), and (1)(1)(1), where 1 is the one-vertex tree, 2 is the two-vertex tree, 3a and 3b are the two three-vertex trees, and parentheses record the partitioning into forests. (1,1)(1) is excluded because the numbers of trees per forest decreases.

MATHEMATICA

m = 20; CoefficientList[Series[Product[1/(1-((1-Sqrt[1-4x])/2)^k), {k, m}], {x, 0, m}], x]

nmax = 30; CoefficientList[Series[1/QPochhammer[(1 - Sqrt[1 - 4*x])/2], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 10 2020 *)

CROSSREFS

Cf. A000041, A000108.

Sequence in context: A071732 A289804 A071736 * A148938 A082306 A124431

Adjacent sequences:  A286952 A286953 A286954 * A286956 A286957 A286958

KEYWORD

nonn

AUTHOR

David Bevan, May 22 2017

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 May 24 14:36 EDT 2022. Contains 354037 sequences. (Running on oeis4.)