The answer is \[ [x^M] \frac{1}{{}_N \mathrm{H}_M}\prod_{i=1}^{N} \left( C_i + (C_i+1)x + (C_i+2)x^2 + \cdots \right). \]We already know\[ 1+x+x^2+\cdots = \frac{1}{1-x}. \]\begin{align} x+2x^2 + 3x^3 + \cdots &= x \frac{d}{dx} (1+x+x^2+\c…