Moments for generalizations of a coin flip game

We derive a recursive formula for the moments of the number of flips using a possibly biased coin to produce a prescribed finite binary string $S$ when $S$ is either a run of heads or a run of heads followed by a tails. Our recursive formula involve certain sums, which we simplify by using a one-parameter extension of the well-studied Eulerian number, which belongs to the two-parameter family of numbers introduced by Graham, Knuth, and Patashnik. We also use the Goulden--Jackson cluster method and Fa\`a di Bruno's formula to establish a closed formula for the moments in a more general situation where a die having an arbitrary number of faces with possibly different probabilities is rolled repeatedly until a prescribed finite word occurs.