Stirling Numbers of the First Kind/Examples/5th Falling Factorial

Example of Stirling Numbers of the First Kind

 * $x^{\underline 5} = x^5 - 10 x^4 + 35 x^3 - 50 x^2 + 24 x$

and so:
 * $\dbinom x 5 = \dfrac 1 {120} \paren {x^5 - 10 x^4 + 35 x^3 - 50 x^2 + 24 x}$

Proof
Follows directly from Stirling's triangle of the first kind (unsigned):

Also from Stirling's triangle of the first kind (signed):

By definition of binomial coefficient:
 * $\dbinom x 5 = \dfrac {x^{\underline 5} } {5!} = \dfrac {x^{\underline 5} } {120}$

Hence:
 * $\dbinom x 5 = \dfrac 1 {120} \paren {x^5 - 10 x^4 + 35 x^3 - 50 x^2 + 24 x}$