%I
%S 1,3,18,215,3600,80136,2213036,73068543,2806959015,123002168300,
%T 6055381161852,330885794632536,19872950226273053,1301261803764756855,
%U 92259974680854975000,7041606755629152575055,575638367425376279620662,50180725346542105445190603
%N Number of binary rooted trees with n leaves of n colors and all nonleaf nodes having outdegree 2.
%C Not all of the n colors need to be used.
%H Alois P. Heinz, <a href="/A319580/b319580.txt">Table of n, a(n) for n = 1..352</a>
%H V. P. Johnson, <a href="http://people.math.sc.edu/czabarka/Theses/JohnsonThesis.pdf">Enumeration Results on Leaf Labeled Trees</a>, Ph. D. Dissertation, Univ. Southern Calif., 2012.
%p A:= proc(n, k) option remember; `if`(n<2, k*n, `if`(n::odd, 0,
%p (t> t*(1t)/2)(A(n/2, k)))+add(A(i, k)*A(ni, k), i=1..n/2))
%p end:
%p a:= n> A(n$2):
%p seq(a(n), n=1..20); # _Alois P. Heinz_, Sep 23 2018
%o (PARI) a(n)={my(v=vector(n)); v[1]=n; for(n=2, n, v[n]=sum(j=1, (n1)\2, v[j]*v[nj]) + if(n%2, 0, binomial(v[n/2]+1, 2))); v[n]} \\ _Andrew Howroyd_, Sep 23 2018
%Y Main diagonal of A319539.
%Y Cf. A319369, A319541.
%K nonn
%O 1,2
%A _Andrew Howroyd_, Sep 23 2018
