We admit that $P \left( e ^ { - t } \right) \sim \sqrt { \frac { t } { 2 \pi } } \exp \left( \frac { \pi ^ { 2 } } { 6 t } \right)$ as $t$ tends to $0 ^ { + }$.
By applying formula (1) to $t = \frac { \pi } { \sqrt { 6 n } }$, prove that
$$p _ { n } \sim \frac { \exp \left( \pi \sqrt { \frac { 2 n } { 3 } } \right) } { 4 \sqrt { 3 } n } \quad \text { as } n \rightarrow + \infty$$