cp's OEIS Frontend

A121686 Number of branches in all binary trees with n edges. A binary tree is a rooted tree in which each vertex has at most two children and each child of a vertex is designated as its left or right child.

%I A121686 #17 Jul 28 2020 11:10:08
%S A121686 2,6,22,84,324,1254,4862,18876,73372,285532,1112412,4338536,16938120,
%T A121686 66192390,258909390,1013586540,3971224620,15571021620,61096813140,
%U A121686 239888764440,942483155640,3705043827420,14573172387852,57351122857944
%N A121686 Number of branches in all binary trees with n edges. A binary tree is a rooted tree in which each vertex has at most two children and each child of a vertex is designated as its left or right child.
%H A121686 Alois P. Heinz, <a href="/A121686/b121686.txt">Table of n, a(n) for n = 1..500</a>
%H A121686 Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a>, 2011. [Cached copy]
%H A121686 Guo-Niu Han, <a href="https://arxiv.org/abs/2006.14070">Enumeration of Standard Puzzles</a>, arXiv:2006.14070 [math.CO], 2020.
%F A121686 a(n) = Sum_{k=1..n} k*A121685(n,k).
%F A121686 G.f.: (1 - 2*z) * (1 - 3*z - (1 - z)*sqrt(1 - 4*z))/(z^2*sqrt(1 - 4*z)).
%F A121686 Recurrence: (n + 2)*(n^2 - 2*n + 3)*a(n) = 2*(2*n - 1)*(n^2 + 2)*a(n-1). - _Vaclav Kotesovec_, Dec 10 2013
%F A121686 a(n) = 2*(n^2 + 2)*binomial(2*n, n)/((n + 1)*(n + 2)). - _Vaclav Kotesovec_, Dec 10 2013
%e A121686 a(1) = 2 because we have two binary trees with 1 edge, namely / and \, with a total of 2 branches.
%p A121686 G:=(1-2*z)*(1-3*z-(1-z)*sqrt(1-4*z))/z^2/sqrt(1-4*z): Gser:=series(G,z=0,31): seq(coeff(Gser,z,n),n=1..27);
%Y A121686 Cf. A121685.
%K A121686 nonn
%O A121686 1,1
%A A121686 _Emeric Deutsch_, Aug 15 2006