The nature of $(4x^3+6x^2+4x+1)^k$

Question

I am trying to identify the mathematical properties of this polynomial $(4x^3+6x^2+4x+1)^k$. Most of all I’ve been attempting to describe an actual relationship to binomial coefficients, such as binomial$(4,n)^k$, and it’s very nature itself. How would this polynomial be written in a ‘concise manner’ resembling the binomial properties instead of just terms? Such as the equivalent of writing binomial$(n,k)=n!/(k!(n-k)!)$...? On Wolfram the gamma function shows up many times, it’s use seems unsatisfactory, and too mathematically rigorous. I’ve tried several methods of finding a useful method for years; multinomials, examining Stirling numbers, Catalan numbers, and so on, and nothing seems to work. I’m not worried about the complexity, it just a matter of finding a better more elegant representation.

Answer 1

Sure. You can use binomial coefficients as the thing is $(x+1)^4 - x^4.$ If you know how to write $(a-b)^k,$ you have a way to write $\left((x+1)^4 - x^4 \right)^k.$