Bell Polynomial

Two different Generating Functions for the Bell polynomials for $n>0$ are given by

$$\phi_n(x) \equiv e^{-x} \sum_{k=0}^\infty {k^n x^k\over k!}$$

or

$$\phi_n(x)=x\sum_{k=1}^n {n-1\choose k-1}\phi_{k-1}(x),$$

where ${n\choose k}$ is a Binomial Coefficient.

The Bell polynomials are defined such that $\phi_n(1)=B_n$, where $B_n$ is a Bell Number. The first few Bell polynomials are

$$\phi_0(x) = 1$$

$$\phi_1(x) = x$$

$$\phi_2(x) = x+x^2$$

$$\phi_3(x) = x+3x^2+x^3$$

$$\phi_4(x) = x+7x^2+6x^3+x^4$$

$$\phi_5(x) = x+15x^2+25x^3+10x^4+x^5$$

$$\phi_6(x) = x+31x^2+90x^3+65x^4+15x^5+x^6.$$

See also Bell Number

References

Bell, E. T. ``Exponential Polynomials.'' Ann. Math. 35, 258-277, 1934.