An explicit bound for the number of partitions into roots

We use the saddle point method to prove an explicit upper bound for the number of representations of a positive integer n   into the form ⌊a1⌋+⌊a2⌋+…+⌊ak⌋, where k   and a1,a2,…,aka1,a2,…,ak are positive integers. We also give an asymptotic formula for this number as n→∞n→∞.