SMO 2010 Senior Round 1 Question 34

Let $P(x)=a_0+a_1x+a_2x^2+\cdots+a_nx^n$ be a polynomial in $x$ where the coefficients $a_0,a_1,a_2,\ldots,a_n$ are non-negative integers. If $P(1)=25$ and $P(27)=1771769$, find the value of $a_0+2a_1+3a_2+\cdots+(n+1)a_n$.