Bell Polynomials

Question

The complete Bell polynomials $B_n(x_1, x_2, \ldots, x_n)$ are defined through the relation $$\sum_{n=0}^{\infty} B_n(x_1, x_2, \ldots, x_n) \frac{t^n}{n!} =\exp\Big( \sum_{n=1}^{\infty} x_n \frac{t^n}{n!}\Big).$$ Is there any known formula for the $B_n(x_1, x_2, \ldots, x_n)$ ? I am looking for an expression $$B_n(x_1, x_2, \ldots , x_n)=\sum_I \alpha_I x_{1}^{i_1}x_2^{i_2}\cdots x_n^{i_n},$$ where the sum is taken over partitions of $n$.

Answer 1

You can write $$ B_n (x_1 ,x_2 , \ldots ,x_n ) = n!\sum\limits_{\substack{i_1 ,i_2 , \ldots ,i_n \ge 0 \\[0.25ex] i_1 + 2i_2 + \ldots + ni_n = n}} {\prod\limits_{j = 1}^n {\frac{{x_j^{i_j } }}{{(j!)^{i_j } \cdot i_j !}}} } . $$ This may be proved by expanding the exponential function in the definition and collecting the coefficients of like powers of $t$.