A214202 - OEIS

%I #16 Jun 18 2017 13:52:58

%S 0,0,0,0,4,0,8,32,104,352,1264,4480,15992,57408,207152,750144,2725456,

%T 9931328,36282464,132852224,487443672,1791742592,6597006896,

%U 24326190016,89825979568,332110462016,1229345599520,4555536068352,16898439030192,62743172964224,233170424975072,867250463225984,3228189434389152,12025362901992064,44827564359795392

%N Number of rooted planar binary unlabeled trees with n leaves and caterpillar index = 4.

%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 + 16*x^4] - Sqrt[1 - 4*x + 32*x^5]) + O[x]^35 // CoefficientList[#, x]& (* _Jean-François Alcover_, Nov 07 2016, after Maple *)

%Y Cf. A025266, A000108, A025271, A214198-A214205.

%K nonn

%O 0,5

%A _N. J. A. Sloane_, Jul 07 2012