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%I #16 Jun 18 2017 13:53:46
%S 0,0,0,2,0,4,12,36,120,392,1288,4284,14304,48024,162024,548872,
%T 1866416,6368464,21797776,74822636,257513344,888439192,3072153864,
%U 10645835384,36964041872,128584760560,448087042160,1564065659608,5467992829120,19144550862960,67123334707984,235658063191312,828405764175712,2915610778184352,10273466501139232,36239527330228044
%N Number of rooted planar binary unlabeled trees with n leaves and caterpillar index = 3.
%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.
%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)*(Sqrt[1-4*x+8*x^3]-Sqrt[1-4*x+16*x^4])+O[x]^36 // 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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