Wedderburn-Etherington number

The $n$th Wedderburn-Etherington number counts how many weakly binary trees can be constructed such that each graph vertex (not counting the root vertex) is adjacent to no more than three other such vertices, for a given number $n$ of nodes. The first few Wedderburn-Etherington numbers are 1, 1, 1, 2, 3, 6, 11, 23, 46, 98, 207, 451, 983, etc. listed in A001190 of Sloane’s OEIS. Michael Somos gives the following recurrence relations:

$$a_{2n}=\frac{1}{2}{a}_n{a}_{n+1}+\sum _{i=1}^n{a}_i{a}_{2n-i}$$

and

$$a_{2n-1}=\sum _{i=0}^{n-1}{a}_{i+1}{a}_{2n-i}$$

with $a_1=a_2=1$ in both relations.