Formulas for the Bell polynomials can be found here:
http://en.wikipedia.org/wiki/Bell_polynomials
and here:
http://mathworld.wolfram.com/BellPolynomial.html
There's a method involving taking a matrix determinant, a convolution formula, and a sum over partitions (we are only interested in the "complete" Bell polynomials \( B_n \) here).
E.g.
\( (x \diam y)_n = \sum_{j=1}^{n-1} {n \choose j} x_j y_{n-j} \)
then
\( B_{n,k}(x_1, ..., x_{n-k+1}) = \frac{x_n^{k \diam}}{k!} \)
(where \( x^{k \diam} \) is like "exponentiating" the sequence with the "diamond" operator)
and
\( B_n(x_1, ..., x_n) = \sum_{k=1}^n B_{n,k}(x_1, ..., x_{n-k+1}) \)
http://en.wikipedia.org/wiki/Bell_polynomials
and here:
http://mathworld.wolfram.com/BellPolynomial.html
There's a method involving taking a matrix determinant, a convolution formula, and a sum over partitions (we are only interested in the "complete" Bell polynomials \( B_n \) here).
E.g.
\( (x \diam y)_n = \sum_{j=1}^{n-1} {n \choose j} x_j y_{n-j} \)
then
\( B_{n,k}(x_1, ..., x_{n-k+1}) = \frac{x_n^{k \diam}}{k!} \)
(where \( x^{k \diam} \) is like "exponentiating" the sequence with the "diamond" operator)
and
\( B_n(x_1, ..., x_n) = \sum_{k=1}^n B_{n,k}(x_1, ..., x_{n-k+1}) \)