Condition for Equality of Adjacent Binomial Coefficients

Theorem
Let $n \in \Z_{>0}$ be a (strictly) positive integer.

Let $\dbinom n k$ denote a binomial coefficient for $k \in \Z$.

Then:
 * $\dbinom n k = \dbinom n {k + 1}$


 * $n$ is an odd integer
 * $k = \dfrac {n - 1} 2$
 * $k = \dfrac {n - 1} 2$

Sufficient Condition
Let $n$ be odd and $k = \dfrac {n - 1} 2$.

Let $n = 2 m + 1$ for some $m \in \Z_{\ge 0}$.

We have:

Hence:

Necessary Condition
Let $n$ and $k$ be such that:


 * $\dbinom n k = \dbinom n {k + 1}$

From Condition for Increasing Binomial Coefficients, it is not the case that:
 * $0 \le k < \dfrac {n - 1} 2$

as under such a condition we would have:


 * $\dbinom n k = \dbinom n {k + 1}$

So:
 * $k \ge \dfrac {n - 1} 2$

By the Symmetry Rule for Binomial Coefficients, we also have that:


 * $\dfrac {n - 1} 2 < k \le n \iff \dbinom n k > \dbinom n {k + 1}$

and so for $\dbinom n k = \dbinom n {k + 1}$ it is not the case that $\dfrac {n - 1} 2 < k \le n$.

So:
 * $k \le \dfrac {n - 1} 2$

Hence that means:
 * $k = \dfrac {n - 1} 2$

which means:
 * $n = 2 k + 1$

and so $n$ is odd.