[bobby07]

I have just found and entry on OEIS for the general formula for the first problem. http://oeis.org/A000055

G.f.: A(x) = 1 + T(x)-T^2(x)/2+T(x^2)/2, where T(x) = x + x^2 + 2*x^3 + … is g.f. for A000081

I havent checked it, I presume its wrong tho?

[Someone]

The original article asks for a closed-form solution. "Closed-form" is not precisely defined (http://en.wikipedia.org/wiki/Closed-form_expression, http://...), but I do not think anybody would call that a closed-form solution.