Given polynomials of the form $$(1+x+x^2+x^3+\cdots+x^k)^n $$ We can calcualte the coefficients by writing it in the form $$(1-x^{k+1})^n \over (1-x)^n$$ and using the power series $(1-x)^{-n}$, as has been explained nicely on SE here and here among other places. Thank you all. Just curious to know where these methods orignated from. Euler seems unaware of it, instead employing a simple but very cumbersome method here.

Also is there any reason to prefer using $$(1-x^k)^n \over (1-x)^n$$ over using the equivalent $$(x^k-1)^n \over (x-1)^n$$ or is it just a convention of sorts that everyone uses the former?

Thank you.

  • $\begingroup$ My personally preference is to go back and forth between $~\displaystyle (1-x^k)^n \over (1-x)^n ~$ and $~\displaystyle (x^k - 1)^n \over (x - 1)^n~$ depending on whether $~x < 1~$ or not. I prefer that the denominator, with the $~n~$ exponent ignored, be non-negative. $\endgroup$ Dec 3, 2023 at 17:34


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