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A007853 Number of maximal antichains in rooted plane trees on n nodes. 15
1, 2, 5, 15, 50, 178, 663, 2553, 10086, 40669, 166752, 693331, 2917088, 12398545, 53164201, 229729439, 999460624, 4374546305, 19250233408, 85120272755, 378021050306, 1685406494673, 7541226435054, 33852474532769, 152415463629568, 688099122024944 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Also the number of initial subtrees (emanating from the root) of rooted plane trees on n vertices, where we require that an initial subtree contains either all or none of the branchings under any given node. The leaves of such a subtree comprise the roots of a corresponding antichain cover. Also, in the (non-commutative) multicategory of free pure multifunctions with one atom, a(n) is the number of composable pairs whose composite has n positions. - Gus Wiseman, Aug 13 2018

LINKS

Table of n, a(n) for n=1..26.

R. Bacher, On generating series of complementary plane trees arXiv:math/0409050 [math.CO], 2004.

M. Klazar, Twelve countings with rooted plane trees, European Journal of Combinatorics 18 (1997), 195-210; Addendum, 18 (1997), 739-740.

Index entries for sequences related to rooted trees

FORMULA

G.f.: (1/4) * (3 - 2*x - sqrt(1-4*x) - sqrt(2) * sqrt((1+2*x) * sqrt(1-4*x) + 1 - 8*x + 2*x^2)) [from Klazar]. - Sean A. Irvine, Feb 06 2018

MATHEMATICA

ie[t_]:=If[Length[t]==0, 1, 1+Product[ie[b], {b, t}]];

allplane[n_]:=If[n==1, {{}}, Join@@Function[c, Tuples[allplane/@c]]/@Join@@Permutations/@IntegerPartitions[n-1]];

Table[Sum[ie[t], {t, allplane[n]}], {n, 9}] (* Gus Wiseman, Aug 13 2018 *)

CROSSREFS

Cf. A000081, A000108, A001003, A001006, A126120, A317713, A318046, A318048, A318049.

Sequence in context: A157135 A196836 A279553 * A149952 A060049 A107590

Adjacent sequences:  A007850 A007851 A007852 * A007854 A007855 A007856

KEYWORD

nonn

AUTHOR

Martin Klazar (klazar(AT)kam.mff.cuni.cz)

EXTENSIONS

More terms from Sean A. Irvine, Feb 06 2018

STATUS

approved

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Last modified October 19 09:21 EDT 2018. Contains 316339 sequences. (Running on oeis4.)