Not getting the output as expected in a Python program that uses dynamic programming

I've written this program that uses dynamic programming:

def count_partitions(n, k):
    if n < k:
        return 0
    elif n == k == 3:
        return 1
    else:
        # Initialize table with base cases
        table = [[0] * (k+1) for _ in range(n+1)]
        table[3][3] = 1

        # Fill in table using recurrence relation
        for i in range(4, n+1):
            for j in range(3, k+1):
                table[i][j] = (j-2) * table[i-1][j] + (j-1) * table[i-1][j-1]

        return table[n][k]

The problem is for the following senario:

A student care center partitions students into groups and supervises them for self-study. However, there is a triumph of triplets from the same family who always talk to one another, center staff will not want to place any two of them in the same group. Suppose there are n students including the triplets, how many ways can the student care center partition these n students into k groups? It is allowed to have only one student in a group, but not allowed to have an empty group. The order of the groups does not matter, so it is considered as the same way if one grouping can be
converted to another by rearranging the sequence of the group.
G(n, k) = G(n-1, k-1) + k × G(n-1, k)

I'm making this program that can be used to solve a specific instance of the partitioning problem by providing the values of "n" and "k" as input.

The "count_partitions" function is used to calculate the number of ways to partition "n" distinct items into "k" non-empty sets.
The "count_partitions" function in the given program also contains a specific case for when both "n" and "k" are equal to 3. In this case, the function returns the value 1, indicating that there is only one way to partition 3 distinct items into 3 non-empty sets.

My code runs, but I don't get the expected results. I would expect these outputs:

# Outputs 1, should output 3
count_partitions(4, 3)
# Outputs 1, should output 9
count_partitions(5, 3)
# Outputs 21, should output 37
count_partitions(6, 4)

Since you haven't provided a description of the problem, we can't help you find the right solution, but we can get you started on understanding why your current program behaves the way it does.

The simplest one is the inputs n=4, k=3, which outputs 1. This is because you initialize the table like this:

    table = [[0] * (k+1) for _ in range(n+1)]
    table[3][3] = 1

So your table, after this executes will be a 5x4 matrix of 0s, except for position [3,3], which is 1:

# k=0 1 2 3
  [ 0 0 0 0 ] # n=0
  [ 0 0 0 0 ] #   1
  [ 0 0 0 0 ] #   2
  [ 0 0 0 1 ] #   3
  [ 0 0 0 0 ] #   4

Then your loops start at i=4 (in the n-axis) and j=3 (in the k-axis). It also stops at i=4, j=3 as well.

for i in range(4, n+1):
    for j in range(3, k+1):

Then you compute a single value (since the loops only run once):

# i=4, j=3
# table[3][3] = 1
# all other table[x][y] = 0

table[i][j] = (j-2) * table[i-1][j] + (j-1) * table[i-1][j-1]

# This reduces to
table[4][3] = 1 * 1 + 2 * 0 = 1

Then you return table[n][k], again with n=4, k=3, so this returns 1, which was set in the loop.

What you need to do is look at what's happening for this example and explain what's supposed to happen instead. Maybe in explaining it, you will find the problem.

Based on your edit, you are trying to implement this recurrence:

G(n, k) = G(n-1, k-1) + k × G(n-1, k)

What you have is

table[i][j] = (j-2) * table[i-1][j] + (j-1) * table[i-1][j-1]

So, to start, you've learned why variable names are important. It's hard to see whether these expressions line up. Let's start by renaming your target recurrence (from G to T, n to i, k to j), and renaming your code from table to t:

T(i, j) = T(i-1, j-1) + j × T(i-1, j)
t[i][j] = (j-2) * t[i-1][j] + (j-1) * t[i-1][j-1]

Now the order is different, too. Let's re-arrange terms and line them up:

T(i, j) = j     × T(i-1, j)   +           T(i-1, j-1)
t[i][j] = (j-2) * t[i-1][j]   +   (j-1) * t[i-1][j-1]

Do you see now what the problem is? The recurrence you've coded isn't the recurrence you're trying to make.