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%I #28 Jul 21 2018 19:22:14
%S 1,3,14,80,510,3479,24848,183465,1389090,10726452,84150858,668825768,
%T 5373971036,43580383095,356234802952,2932097981824,24279982680870,
%U 202134854855973,1690839212784240,14204198452365180,119784707913644598,1013675671656956976
%N Number of non-crossing plants in the (n+1)-sided regular polygon (contains non-crossing trees).
%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 F. Chapoton, F. Hivert, J.-C. Novelli, <a href="http://arxiv.org/abs/1307.0092">A set-operad of formal fractions and dendriform-like sub-operads</a>, arXiv preprint arXiv:1307.0092 [math.CO], 2013.
%H J.-L. Loday, <a href="https://www.emis.de/journals/SLC/wpapers/s53loday.html">Inversion of integral series enumerating planar trees</a>, Séminaire Lotharingien de Combinatoire 53, Art. B53d, 16 pp, 2005. See example (g).
%F x = (y - y^2 - y^3)/(1+y)^2 is the inverse of the generating function.
%F a(n) = sum(j=0..n-1, (sum(i=0..n-j-1, binomial(i+n-1,n-1) *binomial(i+n,n-j-i-1))) *binomial(n,j))/n, n>0, a(0)=0. - _Vladimir Kruchinin_, Oct 12 2011
%e a(2)=3 because the non-crossing plants in a triangle are the three non-crossing trees, made of two sides.
%e G.f. = x + 3*x^2 + 14*x^3 + 80*x^4 + 510*x^5 + 3479*x^6 + 25848*x^7 + ...
%t a[n_] := If[n==0, 0, Sum[Sum[Binomial[i+n-1, n-1]*Binomial[i+n, n-j-i-1], {i, 0, n-j-1}]*Binomial[n, j], {j, 0, n-1}]/n]; Table[a[n], {n, 1, 16}] (* _Jean-François Alcover_, Feb 20 2017, after _Vladimir Kruchinin_ *)
%o (Maxima)
%o a(n):=if n=0 then 0 else sum((sum(binomial(i+n-1,n-1)*binomial(i+n,n-j-i-1),i,0,n-j-1))*binomial(n,j),j,0,n-1)/n; /* _Vladimir Kruchinin_, Oct 12 2011 */
%o (PARI) {a(n) = if( n<1, 0, polcoeff( serreverse( (x - x^2 + x^3) / (1 + x)^2 + x * O(x^n)), n))}; /* _Michael Somos_, Dec 31 2014 */
%Y Cf. A006013.
%K nonn
%O 1,2
%A _F. Chapoton_, Aug 31 2006
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