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A007853 Number of maximal antichains in rooted plane trees on n nodes. 15

%I

%S 1,2,5,15,50,178,663,2553,10086,40669,166752,693331,2917088,12398545,

%T 53164201,229729439,999460624,4374546305,19250233408,85120272755,

%U 378021050306,1685406494673,7541226435054,33852474532769,152415463629568,688099122024944

%N Number of maximal antichains in rooted plane trees on n nodes.

%C 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

%H R. Bacher, <a href="http://arxiv.org/abs/math/0409050">On generating series of complementary plane trees</a> arXiv:math/0409050 [math.CO], 2004.

%H M. Klazar, <a href="http://dx.doi.org/10.1006/eujc.1995.0095">Twelve countings with rooted plane trees</a>, European Journal of Combinatorics 18 (1997), 195-210; Addendum, 18 (1997), 739-740.

%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>

%F 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

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

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

%t Table[Sum[ie[t],{t,allplane[n]}],{n,9}] (* _Gus Wiseman_, Aug 13 2018 *)

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

%K nonn

%O 1,2

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

%E More terms from _Sean A. Irvine_, Feb 06 2018

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Last modified November 16 22:10 EST 2018. Contains 317275 sequences. (Running on oeis4.)