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%I #19 Jun 18 2017 13:52:45
%S 0,1,1,2,5,14,26,100,333,1110,3742,12764,44258,154636,544660,1932360,
%T 6900029,24780390,89445174,324326060,1180834390,4315287140,
%U 15823305516,58200045432,214672363410,793883691004,2942917457772,10933569255832,40704185771812,151826357818840,567322837830824,2123429246035600,7960199797453213,29884582184913542
%N Number of rooted planar binary unlabeled trees with n leaves and caterpillar index <= 5.
%H Filippo Disanto, <a href="http://arxiv.org/abs/1202.5668">The size of the biggest Caterpillar subtree in binary rooted planar trees</a>, arXiv preprint arXiv:1202.5668 [math.CO], 2012.
%H Filippo Disanto, <a href="http://www.kurims.kyoto-u.ac.jp/EMIS/journals/SLC/wpapers/s68disanto.html">Unbalanced subtrees in binary rooted ordered and un-ordered trees</a>, Séminaire Lotharingien de Combinatoire, 68 (2013), Article B68b.
%p C:=(1-sqrt(1-4*x))/2; # A000108 with a different offset
%p # F-(k): gives A025266, A025271, A214200, A214203
%p Fm:=k->(1/2)*(1-sqrt(1-4*x+2^(k+1)*x^(k+1)));
%p Sm:=k->seriestolist(series(Fm(k),x,50));
%p # F+(k): gives A000108, A214198, A214201, A214204
%p Fp:=k->C-Fm(k-1);
%p Sp:=k->seriestolist(series(Fp(k),x,50));
%p # F(k): gives A025266, A214199, A214202, A214205
%p F:=k->Fm(k)-Fm(k-1);
%p S:=k->seriestolist(series(F(k),x,50));
%t (1/2)*(1 - Sqrt[1 - 4*x + 64*x^6]) + O[x]^34 // CoefficientList[#, x]& (* _Jean-François Alcover_, Nov 07 2016, after Maple *)
%Y Cf. A025266, A000108, A025271, A214198-A214205.
%K nonn
%O 0,4
%A _N. J. A. Sloane_, Jul 07 2012
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