%I A120989
%S A120989 2,9,34,123,440,1573,5642,20332,73644,268090,980628,3603065,13293540,
%T A120989 49234605,182991450,682341000,2551955340,9570762990,35985909180,
%U A120989 135628219350,512302356384,1939078493154,7353556121924,27936898370248
%N A120989 Level of the first leaf (in preorder traversal) of a binary tree, summed 
               over 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.
%C A120989 a(n)=Sum(k*A120988(n,k),k=1..n).
%F A120989 a(n)=2n(7n+13)binomial(2n+1,n)/[(n+2)(n+3)(n+4)]. G.f.=z(1+C)C^4, where 
               C=[1-sqrt(1-4z)]/(2z) is the Catalan function. G.f.=2[1+2z-sqrt(1-4z)]/
               [1-2z+sqrt(1-4z)]^2.
%e A120989 a(1)=2 because for each of the trees / and \ the level of the first leaf 
               is 1.
%p A120989 a:=n->2*n*(7*n+13)*binomial(2*n+1,n)/(n+2)/(n+3)/(n+4): seq(a(n),n=1..27);
%Y A120989 Cf. A120988.
%Y A120989 Sequence in context: A150936 A109719 A000524 this_sequence A010763 A077234 
               A091526
%Y A120989 Adjacent sequences: A120986 A120987 A120988 this_sequence A120990 A120991 
               A120992
%K A120989 nonn
%O A120989 1,1
%A A120989 Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 30 2006