The a(n) elements of L of length 2n+2 are the sequences of parentheses
that occur in fully parenthesized (n+2)-term expressions. E.g. () is the
 parentheses sequence for (ab); (()) is the parentheses sequence for ((ab)c)
and (a(bc)); ((())) and (()()) are the parentheses sequences for (((ab)c)d), ((a(bc))d),
((ab)(cd)), (a((bc)d)), (a(b(cd))). If the outermost pair of parentheses
of an element of L of length 2n+2 is stripped off, the resulting string
of length 2n is a properly nested string of parentheses. As is well known,
the properly nested parentheses of length 2n are enumerated by the Catalan
number C(n), confirming that a(n) <= C(n). Note that the fully parenthesized 
(n+2)-term expressions are also enumerated by a Catalan number, in this
case by C(n+1). For n >= 3, a(n) is strictly less than C(n): the parentheses
sequences in L of length 2n+2 are restricted to those with the property that
any substring immediately enclosed by a pair of matching parentheses
has at most two returns.