I was reading "A NOTE ON THE INVERSION OF POWER SERIES" and was able to follow the paper's reasoning until the bottom of the second page, where it states:
in fact we can calculate the power series $y^n$ for any real $n$ by making use of the binomial expansion $y^n=[a_0+a_1x+\cdots+a_mx^m+\cdots]^n\\a_0^n\left[1+v\right]^n=a_0^n\left[1+nv+\cdots+\binom{n}{r}v^r+\cdots\right].$
I understand that he simply substituted for y in the first equation and then expanded using the binomial theorem in the second equation. What I do not understand is how the two equations are related. I assumed it would be explained later, so I continued to read, and it was relatively explained.
Now we construct a table as in figure 2 for the function v. Above the top row, insert the numbers $\binom{n}{r}$ corresponding to the r-th column. Then the coefficient of $x^m$ in $y^n$ is $a_0^n\left[1+w_{m,n}\right]$ where $w_{m,n}=\sum\binom{n}{r}a_{m,r}$ is the accumulated product of the terms in the m-th row with the corresponding numbers $\binom{n}{r}$. Note that each sum is a finite sum because v has no constant term and therefore $a_{m,r}=0$ for $r>m$.
What I still do not understand is how and why that is true in regards to the coefficient of $x^m$. I would greatly appreciate any help in deducing the steps to derive the author's conclusion.