Maximising a weighted product

Let $S_m = (x_1, x_2, ... , x_m)$ be the $m$-tuple of positive real numbers with $x_1 + x_2 + ... + x_m = m$ for which $P_m = x_1 * x_2^2 * ... * x_m^m$ is maximised.

For example, it can be verified that $\lfloor P_{10}\rfloor = 4112$ ($\lfloor \rfloor$ is the integer part function).

Find $\sum \lfloor P_m\rfloor$ for $2 ≤ m ≤ 15$.