The full state vector of boson sampling is generated by passing S single photons through beam splitters of M modes. We express the initial Fock state in terms of 2^S−1 generalized coherent states, making possible the exact application of the unitary evolution. Due to the favorable polynomial scaling of numerical effort in M, we can investigate Rényi entanglement entropies for moderate particle and huge mode numbers. We find symmetric Page curves with a maximum entropy at equal partition, which is almost independent on Rényi index. Furthermore, the maximum entropy as a function of mode index saturates for M ≥ S² in the collision-free subspace case. The asymptotic value of the entropy increases linearly with S. In addition, we show that the build-up of the entanglement leads to a cusp in the asymmetric entanglement curve. Maximum entanglement is reached well before the mode population is distributed over the whole system.