126. Binomial coefficients. Find the largest value of $y$ such that in the expansion of some binomial, $y$ consecutive coefficients are in the ratio $1: 2: 3: \ldots: y$. Determine this expansion and list the corresponding coefficients.
126. For the condition of the problem to be satisfied at $y=3$, for some $n$ and $k$, the equalities $2 C_{n}^{k}=C_{n}^{k+1}$ and $3 C_{n}^{k}=C_{n}^{k+2}$ must hold. After simplification, we get $n=3 k+2$ and $3(k+1)(k+2)=$ $(n-k)(n-k-1)$. Solving this system of equations and discarding the negative solution, we get $k=4, n=14$.

Thus, the degree of the corresponding binomial is 14, and the fifth, sixth, and seventh coefficients are 1001, 2002, and 3003, respectively. Since for $y=3$ the solution is unique, there are no solutions for $y>3$.