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Let $B_{n,k}$ and $A_{n}=\sum_{j=1}^{n}B_{n,j}$ with $A_0=1$ be, respectively, the $(n,k)^{\rm th}$ partial and the $n^{\rm th}$ complete Bell polynomials with indeterminate arguments $x_1,x_2,\ldots$. Congruences for $A_{n}$ and $B_{n,k}$ with respect to a prime number have been studied by several authors. In the present paper, we propose some results involving congruences for $B_{n,k}$ when the arguments are integers. We give a relation between Bell polynomials and we apply it to several congruences. The obtained congruences are connected to binomial coefficients.

Jeffrey Shallit 2009-05-12