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%I #39 Nov 07 2019 19:37:11
%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
%C The g.f. is denoted by y_2 in Bacher 2004 Proposition 7.5 on page 20. - _Michael Somos_, Nov 07 2019
%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
%F a(n) = (1/(n+1))*C(2*n,n) + Sum_{k=0..n-1} ((k+2)/(n+1))*C(2*n-k-1,n-k-1)*Sum_{i=0..floor(k/2)} C(2*i,i)*C(k+i,3*i)/(i+1). - _Vladimir Kruchinin_, Apr 05 2019
%F Given the g.f. A(x) and the g.f. of A213705 B(x), then -x = A(-B(x)). - _Michael Somos_, Nov 07 2019
%e G.f. = x + 2*x^2 + 5*x^3 + 15*x^4 + 50*x^5 + 178*x^6 + 663*x^7 + 2553*x^8 + ... - _Michael Somos_, Nov 07 2019
%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 *)
%o (Maxima)
%o a(n):=1/(n+1)*binomial(2*n,n)+sum((k+2)/(n+1)*binomial(2*n-k-1,n-k-1)*(sum(((binomial(2*i,i))*(binomial(k+i,3*i)))/(i+1),i,0,floor(k/2))),k,0,n-1); /* _Vladimir Kruchinin_, Apr 05 2019 */
%o (PARI) {a(n) = my(A); if( n<0, 0, A = sqrt(1 - 4*x + x * O(x^n)); polcoeff( (3 - 2*x - A - sqrt(2 - 16*x + 4*x^2 + (2 + 4*x) * A)) / 4, n))}; /* _Michael Somos_, Nov 07 2019 */
%Y Cf. A000081, A000108, A001003, A001006, A126120, A213705, A317713, A318046, A318048, A318049.
%K nonn
%O 1,2
%A _Martin Klazar_
%E More terms from _Sean A. Irvine_, Feb 06 2018
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