Maximum gap between binomial coefficients

I recently stumbled on a formula for the largest gap between consecutive items in a row of Pascal’s triangle.

For n ≥ 2,

\max_{1 \leq k \leq n} \left| {n \choose k} - {n \choose k-1}\right| = {n \choose \lfloor \tau \rfloor} - {n \choose \lfloor \tau - 1\rfloor}

where

\tau = \frac{n + 2 - \sqrt{n+2}}{2}

For example, consider the 6th row of Pascal’s triangle, the coefficients of (x + y)6.

1, 6, 15, 20, 15, 6, 1

The largest gap is 9, the gap between 6 and 15 on either side. In our formula n = 6 and so

τ = (8 – √8)/2 = 2.5858

and so the floor of τ is 2. The equation above says the maximum gap should be between the binomial coefficients with k = 2 and 1, i.e. between 15 and 6, as we expected.

I’ve needed a result like this in the past, but I cannot remember now why. I’m posting it here for my future reference and for the reference of anyone else who might need this. I intend to update this post if I run across an application.

More on Pascal’s triangle

Source: Zun Shan and Edward T. H. Wang. The Gaps Between Consecutive Binomial Coefficients. Mathematics Magazine, Vol. 63, No. 2 (Apr., 1990), pp. 122–124

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