# 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 '19 at 7:39

## 1 Answer

first, it is not a wave, wich implies Space and time, it is an oscillation, and you are right, it would come from Doppelereffekt. The frequency yo would hier from a sound source moving away from you and acellarating, would give something like your picture, or a little different if you move accelerating away from a surge with a single frequency. the frequency you hear in the first case ist f=f_0/(1-c/v) c=velocity of sound, v=velocity of car, if now v=at with a acceleration the frequency decreases in time, if v is negativ. the curve would be Asin(f_0*t/(1-at/c)) in the second case f=f_o(1+v/c) again with v=-at but with at

• I am aware of the math and the physics of Dopper Shift but I can't see how your answer addresses the two questions asked in the post? Jul 25 '19 at 12:25