We define a dynamic programming approach where we use a 2D DP table where dp[i][j] means the number of ways to form a playlist of length j using i distinct songs. We will eventually find dp[n][goal].

The key recurrence relations are:

• If we choose a fresh song, there are (n - i + 1) songs available, hence dp[i][j] += dp[i - 1][j - 1] * (n - i + 1)
• If we re-use one of the i songs, considering k songs must be between repeats, we have: dp[i][j] += dp[i][j - 1] * (i - k), where i > k

This draws from the constraints on repeats and expands the playlist iteratively.

This approach uses memoization to recursively compute the number of playlists. It calculates the playlists by considering whether a new song is added or an existing song is reused, accounting for k intervening songs.