Simplifying $ \sum_{1\cdot m_1 + 2\cdot m_2 + \cdots + n \cdot m_n = n} \frac{1}{m_1 ! m_2 ! \cdots m_n !} t^{m_1 + \cdots + m_n} $

Question

Does anybody know how to simplify the expression like the following?

$$ \sum_{1\cdot m_1 + 2\cdot m_2 + \cdots + n \cdot m_n = n} \frac{1}{m_1 ! m_2 ! \cdots m_n !} t^{m_1 + \cdots + m_n} $$

This looks unusual to me since the summation is usually taken over $n$-tuples satisfying $m_1 + \cdots + m_n = n$.