Number base and binomial coefficient

$\overline{C_{0}^{0} } _{x} + \overline{C_{0}^{1} C_{1}^{1}} _{x} + \overline{C_{0}^{2} C_{1}^{2} C_{2}^{2}} _{x} + \dots + \overline{C_{0}^{2016} \dots C_{2016}^{2016}}_{x}$

$= \overline{1} _{x} + \overline{11} _{x} + \overline{121} _{x} + \overline{1331} _{x} + \dots + \overline{C_{0}^{2016} \dots C_{2016}^{2016}}_{x}$

$= 1 +(x+1)+(x^2+2x+1)+(x^3+3x^2+3x^2+1)+ \dots +(x^{2016}+2016x^{2015}+\dots+2016x+1)$

$= 1+(x+1)+(x+1)^2+(x+1)^3+ \dots+ (x+1)^{2016}$

$=\underbrace{\overline{111\dots111}_{(x+1)}}_{2017 \ 1's}$

The calculation above shows that we can put any value to satisfy the equation but it does not .

Considering the boundary of the number base , the value of each single digit must be smaller than the value of number base and number base must be positive integer .

e.g. $\large \overline{2016}_{6}$ is an incorrect expression but $\large \overline{2016}_{7}$ is a correct expression .

Therefore , we need to find the maximum value of the digit which exists in the equation .

Finding patterns , the maximum value of $\large C_{k}^{n}$ is $\large C_{\frac{n}{2}}^{n}$ ( for even numbers) . $\large C_{1008}^{2016}$ is biggest and exists in the equation . $\large C_{1008}^{2016}$ is the binomial coefficient in the expansion of $(x+1)^{2016}$ .

The value of $x$ cannot be smaller than and equal to $\large C_{1008}^{2016}$ .Then , $\large C_{1008}^{2016}+1$ is the minimum possible value of $x$ which satisfies the equation above .