A214203 Number of rooted planar binary unlabeled trees with n leaves and caterpillar index <= 5.

0, 1, 1, 2, 5, 14, 26, 100, 333, 1110, 3742, 12764, 44258, 154636, 544660, 1932360, 6900029, 24780390, 89445174, 324326060, 1180834390, 4315287140, 15823305516, 58200045432, 214672363410, 793883691004, 2942917457772, 10933569255832, 40704185771812, 151826357818840, 567322837830824, 2123429246035600, 7960199797453213, 29884582184913542

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.
• Filippo Disanto, Unbalanced subtrees in binary rooted ordered and un-ordered trees, Séminaire Lotharingien de Combinatoire, 68 (2013), Article B68b.

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)*(1 - Sqrt[1 - 4*x + 64*x^6]) + O[x]^34 // CoefficientList[#, x]& (* Jean-François Alcover, Nov 07 2016, after Maple *)