A214199 Number of rooted planar binary unlabeled trees with n leaves and caterpillar index = 3.

0, 0, 0, 2, 0, 4, 12, 36, 120, 392, 1288, 4284, 14304, 48024, 162024, 548872, 1866416, 6368464, 21797776, 74822636, 257513344, 888439192, 3072153864, 10645835384, 36964041872, 128584760560, 448087042160, 1564065659608, 5467992829120, 19144550862960, 67123334707984, 235658063191312, 828405764175712, 2915610778184352, 10273466501139232, 36239527330228044

Offset: 0

Author

N. J. A. Sloane, Jul 07 2012

Keywords

• nonn

Links

• Filippo Disanto, The size of the biggest Caterpillar subtree in binary rooted planar trees, arXiv preprint arXiv:1202.5668 [math.CO], 2012.

Crossrefs

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

Programs

Maple

C:=(1-sqrt(1-4*x))/2; # A000108 with a different offset
# F-(k): gives A025266, A025271, A214200, A214203
Fm:=k->(1/2)*(1-sqrt(1-4*x+2^(k+1)*x^(k+1)));
Sm:=k->seriestolist(series(Fm(k),x,50));
# F+(k): gives A000108, A214198, A214201, A214204
Fp:=k->C-Fm(k-1);
Sp:=k->seriestolist(series(Fp(k),x,50));
# F(k): gives A025266, A214199, A214202, A214205
F:=k->Fm(k)-Fm(k-1);
S:=k->seriestolist(series(F(k),x,50));

Mathematica

(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 *)