|
%I #29 Aug 04 2019 18:22:23
%S 1,3,12,52,236,1109,5366,26639,135300,701269,3700400,19834973,
%T 107784622,592705377,3292970302,18458954896,104276682820,593056996445,
%U 3392898090908,19512100041995,112729617387020,653965783541960,3807766434556940
%N Subtrees in rooted plane trees on n nodes.
%H Vincenzo Librandi, <a href="/A007856/b007856.txt">Table of n, a(n) for n = 1..1000</a>
%H G.-S. Cheon, H. Kim, L. W. Shapiro, <a href="http://arxiv.org/abs/1410.1249">Mutation effects in ordered trees</a>, arXiv preprint arXiv:1410.1249 [math.CO], 2014.
%H M. Klazar, <a href="http://dx.doi.org/10.1006/eujc.1995.0095">Twelve countings with rooted plane trees</a>, European Journal of Combinatorics 18 (1997), 195-210; Addendum, 18 (1997), 739-740.
%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F 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
%F a(n) = ((n - 1)/2)*CatalanNumber(n-1)*(1 - hypergeom([-1/2, -n], [n - 1], -4)). - _Peter Luschny_, Aug 04 2019
%t 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 *)
%t 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 *)
%K nonn
%O 1,2
%A _Martin Klazar_
%E More terms from _Sean A. Irvine_, Feb 06 2018
|
