A007856 Subtrees in rooted plane trees on n nodes.

1, 3, 12, 52, 236, 1109, 5366, 26639, 135300, 701269, 3700400, 19834973, 107784622, 592705377, 3292970302, 18458954896, 104276682820, 593056996445, 3392898090908, 19512100041995, 112729617387020, 653965783541960, 3807766434556940

Offset: 1

Author

Martin Klazar

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• G.-S. Cheon, H. Kim, L. W. Shapiro, Mutation effects in ordered trees, arXiv preprint arXiv:1410.1249 [math.CO], 2014.
• M. Klazar, Twelve countings with rooted plane trees, European Journal of Combinatorics 18 (1997), 195-210; Addendum, 18 (1997), 739-740.
• Index entries for sequences related to rooted trees

Programs

Mathematica

Rest[CoefficientList[Series[(1/8) (1 + 1/Sqrt[1 - 4  x]) (1 + Sqrt[1 - 4  x] - Sqrt[2]  Sqrt[1 - 10  x + Sqrt[1 - 4  x]]), {x, 0, 33}], x]] (* Vincenzo Librandi, Feb 07 2018 *)
A007856[n_] := ((n-1)/2) CatalanNumber[n-1](1 - Hypergeometric2F1[-1/2, - n, n-1, -4]); Table[A007856[n], {n, 1, 23}] (* Peter Luschny, Aug 04 2019 *)

Formula

G.f.: (1/8) * (1 + 1/sqrt(1-4*x)) * (1 + sqrt(1-4*x) - sqrt(2) * sqrt(1 - 10*x + sqrt(1-4*x))), see Klazar's paper. - Sean A. Irvine, Feb 06 2018

a(n) = ((n - 1)/2)*CatalanNumber(n-1)*(1 - hypergeom([-1/2, -n], [n - 1], -4)). - Peter Luschny, Aug 04 2019

Extensions

More terms from Sean A. Irvine, Feb 06 2018