In this problem, we want to find the following: - The number of ways to decompose \(K\) into \(N\) numbers, each of which is \(1\) or \(\frac{1}{2}\) or \(\frac{1}{4}\) or\(\dots\)

Let us divide these decompositions into two categories: decompositions using \(1\) at least once and decompositions not using \(1\).

Decompositions using \(1\):

The number of such decompositions is equal to the following:

• The number of ways to decompose \(K-1\) into \(N-1\) numbers, each of which is \(1\) or \(\frac{1}{2}\) or \(\frac{1}{4}\) or\(\dots\)

Decompositions not using \(1\):

The number of such decompositions is equal to the following:

• The number of ways to decompose \(K\) into \(N\) numbers, each of which is \(\frac{1}{2}\) or \(\frac{1}{4}\) or\(\dots\)

By multiplying all values by \(2\), we can see that this is equal to the following:

• The number of ways to decompose \(2K\) into \(N-1\) numbers, each of which is \(\frac{1}{2}\) or \(\frac{1}{4}\) or\(\dots\)

Thus, we have dp[N][K] = dp[N-1][K-1] + dp[N][2K], where dp[N][K] is the answer to the problem.

Note that, from the definition of the problem, we have dp[N][K] = 0 for \(N < K\), and we can solve the problem in \(O(N^2)\) time.