Let G(x) be the ordinary (or did you mean exponential?) generating function of a(n) -- i.e., G(x) = sum a(n)x^n (from n = 0 to oo). Then

G(x) = 1 + sum a(n+1)x^(n+1) = 1 + sum (n+2)a(n)x^(n+1) = 1 + d/dx sum a(n)x^(n+2) = 1 + d/dx [ x^2G(x) ] = 1 + 2xG(x) + x^2G'(x)

so we have to solve the first-order linear ordinary differential equation y = 1 + 2xy + x^2y' with y(0) = 1. While doing so is standard (integrating factor) the solution is ugly, involving an exponential integral. The exponential generating function seems as tho it'll be even uglier, however. Am I missing something?