# Is there an explanation for this unexpected similarity between binomial coefficients and waves?

Background

Binomial coefficients appeal mostly in probability, combinatorics number theory etc so were were surprised when we observed something that appeared to belong more to physics than pure mathematics. This question arose out of a problem that we were working on Mathematics Stack Exchange and we observed things that looked like propagation of a wave with fixed amplitude and decreasing frequency. So we are posting this question in Physics community asking if there is something similar to this in Physics.

Theory

Let $$s_{n,a}$$ be the sum of the squares of the multiple of the squares of binomial coefficients as defined below. $$s_{n,a} = \sum_{1\leq \lfloor ak^2 \rfloor\leq n}{n\choose \lfloor ak^2 \rfloor}= {n\choose \lfloor 1^2 a \rfloor} + {n\choose \lfloor 2^2 a \rfloor} + \cdots + {n\choose \lfloor r^2 a \rfloor}.$$

We obtained series expansion $$\frac{S_{n,a}\sqrt{2an}}{2^n}=1+\sum_{k=1}^{\infty}2 e^{-\frac{\pi^2}{4a} k^2} \cos2\pi k x_n+O\left(\frac{\log^3 n}{\sqrt n}\right),$$ where $$x_n$$ is defined in the above link.

Observation

The graph of $$\dfrac{S_{n,a}\sqrt{n}}{2^n}$$ has the shape as shown below

Questions

The graph looks like a wave with fixed amplitude and decreasing frequency, somewhat like frequency modulation where you are only allowed to decrease the frequency but never increase it back. Mathematically, this is due to the $$\cos$$ term in the series expansion.

1. Where in physics do we observe something like this with fixed amplitude and decreasing frequency? May be something like Doppler shift?
2. What is the equation of a wave with such characteristics?
• Consider to make post self-contained and define $x_n$ directly. Jul 25, 2019 at 7:39