CZ-SK-MO 2017 Problem 4

For each sequence of $n$ zeros and $n$ units, we assign a number that is a number sections of the same digits in it. (For example, sequence $00111001$ has $4$ such sections $00, 111,00, 1$ .) For a given $n$ we sum up all the numbers assigned to each such sequence. Prove that the sum total is equal to $(n+1){2n \choose n}$