Negated Upper Index of Binomial Coefficient/Corollary 2
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Corollary to Negated Upper Index of Binomial Coefficient
Let $n, m \in \Z$.
Then:
- $\dbinom n m = \paren {-1}^{n - m} \dbinom {-\paren {m + 1} } {n - m}$
where $\dbinom n m$ is a binomial coefficient.
Proof
\(\ds \dbinom r k\) | \(=\) | \(\ds \paren {-1}^k \dbinom {k - r - 1} k\) | Negated Upper Index of Binomial Coefficient | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \dbinom n {n - m}\) | \(=\) | \(\ds \paren {-1}^{n - m} \dbinom {\paren {n - m} - n - 1} {n - m}\) | setting $r = n$ and $k = n - m$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \dbinom n m\) | \(=\) | \(\ds \paren {-1}^{n - m} \dbinom {-m - 1} {n - m}\) | Symmetry Rule for Binomial Coefficients | ||||||||||
\(\ds \) | \(=\) | \(\ds \paren {-1}^{n - m} \dbinom {- \paren {m + 1} } {n - m}\) |
$\blacksquare$
Sources
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: $(19)$