Estimating a cumulative sum

In this post I mentioned two series which I denoted t(n) and c(n). The former is the number of unlabeled rooted trees with n nodes. The latter is the cumulative sum of the former, i.e. The sequence c(n) is also the number of constraints on an n-step Runge-Kutta method; that s how I became interested in it. Now the t(n) sequence [ ]