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Simplified $\sum_{n=a_1k_1+a_2k_2+\cdots+a_mk_m}\frac{(a_1+a_1+\cdots+a_m)!}{a_1!\cdot a_2!\cdots a_m!}\prod_{i=1}^m \left(f(k_i)\right)^{a_i}$
I'm studying a function of the form
$$b_n=\sum_{n=a_1k_1+a_2k_2+\cdots+a_mk_m}\frac{(a_1+a_2+\cdots+a_m)!}{a_1!\cdot a_2!\cdots a_m!}\prod_{i=1}^m \left(f(k_i)\right)^{a_i}$$
Where the sum is over ...
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