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 constrains on an n-step Runge-Kutta method; that’s how I became interested in it. Now the t(n) sequence has been cataloged as OEIS A000081 and OEIS gives the asymptotic estimate of t(n) for large n as where C = 0.4399… and α = 2.9557…. The cumulative sum of t(n), what I’ve called c(n), is also cataloged in OEIS, sequence number A087803. However, OEIS does not give an asymptotic estimate for this sequence. I’ll give one here. The basis for my derivation is to assume the cumulative sum of the asymptotic estimates gives an asymptotic estimate of the cumulative sum. This is justified by the fact that the sequence is increasing rapidly and only the last few terms contribute much relatively to the sum. Here’s code to visualize the rate of convergence. import numpy as np…