If n be a positive integer number and the variable x less than 1/n, the rational fraction 1/[(1 - x) (1 – 2 x)... (1 - n x)] can be expanded into the series $${}^{ - n}{A_0} + {}^{ - n}{A_1}x + {}^{ - n}{A_2}{x^2} + {}^{ - n}{A_3}{x^3} + \cdot \cdot \cdot$$ and it will be readily seen that the general coefficient -n A m is a positive integer number2). From $$\left( {\frac{d} {{dx}} - n} \right)\frac{{{{\left( {{e^x} - 1} \right)}^n}}} {{n!}} = \frac{{{{\left( {{e^x} - 1} \right)}^{n - 1}}}} {{\left( {n - 1} \right)!}}$$ we may then infer $$\frac{{{{\left( {{e^x} - 1} \right)}^n}}} {{n!}} = \frac{1} {{\left( {\frac{d} {{dx}} - 1} \right)\left( {\frac{d} {{dx}} - 2} \right)...\left( {\frac{d} {{dx}} - n} \right)}} \cdot 1 = \sum\limits_{\lambda = 0}^{\lambda = \infty } {{}^{ - n}{A_\lambda }{{\left( {\frac{d} {{dx}}} \right)}^{ - n - \lambda }} \cdot 1 = \sum\limits_{\lambda = 0}^{\lambda = \infty } {{}^{ - n}{A_\lambda }\frac{{{x^{n + 1}}}} {{\left( {n + \lambda } \right)!}}} }$$ whence $$\sum\limits_{m = 0}^{m = n} {\frac{{{{\left( { - 1} \right)}^{n - m}}}} {{m!\left( {n - m} \right)!}}{e^{mx}} = \sum\limits_{\lambda = 0}^{\lambda = \infty } {{}^{ - n}{A_\lambda }\frac{{{x^{n + \lambda }}}} {{\left( {n + \lambda } \right)!}}} }$$ therefore $$n!{}^{ - n}{A_{m - n}} = \sum\limits_{\lambda = 0}^{\lambda = m} {{{\left( { - 1} \right)}^{n - \lambda }}\left( \begin{gathered} n \hfill \\ \lambda \hfill \\ \end{gathered} \right){\lambda ^m}}$$ which expression is zero for m = 0, 1, 2, 3, ..., n - 1 and n! for m = n, and might also have been obtained by the well known method of decomposition of a rational fraction.