Supposing that we want $k$ distinct integers that sum to at most
$n$ we first choose the minimum value
$$\frac{z}{1-z}$$
and then we choose $k-1$ differences between consecutive values
$$\left(\frac{z}{1-z}\right)^{k-1}.$$
The coefficient $[z^n]$ of the product
$$F(z) = \left(\frac{z}{1-z}\right)^k$$
then gives the number of sets of $k$ distinct positive integers
with the largest element being $n.$ We want the sum however.
Now the minimum value contributes $k$ times, the first difference
$k-1$ and so on, the last difference contributes once.
This gives the product
$$G(z) = \prod_{q=0}^{k-1} \frac{z^{k-q}}{1-z^{k-q}}
= \prod_{q=1}^k \frac{z^q}{1-z^q}.$$
This coefficient gives the set of $k$ distinct positive integers that
sum to exactly $n.$ If we require at most $n$, we sum by multiplying by
$1/(1-z)$ and obtain
$$H(z) = \frac{1}{1-z} \prod_{q=1}^k \frac{z^q}{1-z^q}$$
We may write
$$H(z) = \frac{z^{\frac{1}{2} k(k+1)}}{1-z}
\prod_{q=1}^k \frac{1}{1-z^q}
= z^{\frac{1}{2} k(k+1)} J(z)$$
so that
$$J(z) = \frac{1}{1-z} \prod_{q=1}^k \frac{1}{1-z^q}
\quad\text{and}\quad
H_{n,k} = J_{n-\frac{1}{2} k(k+1), k}.$$
The base case here is $H_{n,k} = 0$ if $n\lt \frac{1}{2} k(k+1)$
and $H_{\frac{1}{2} k(k+1), k} = 1.$
Differentiate to get
$$J'(z) = \frac{1}{1-z} \prod_{q=1}^k \frac{1}{1-z^q}
\left[ \frac{1}{1-z}
+ \frac{1}{z} \sum_{q=1}^k \frac{q z^q}{1-z^q}\right].$$
Extract coefficients on $[z^{n-1}]$ to get
$$n J_{n,k} = \sum_{p=0}^{n-1} J_{p,k}
+ \sum_{q=1}^k q \sum_{p=1}^{\lfloor n/q \rfloor}
J_{n-pq,k}.$$
We find the recurrence
$$\bbox[5px,border:2px solid #00A000]{
J_{n,k} = \frac{1}{n}
\left[ \sum_{p=0}^{n-1} J_{p,k}
+ \sum_{q=1}^k q \sum_{p=1}^{\lfloor n/q \rfloor}
J_{n-pq,k}\right]}$$
where $n\ge 1$ and $J_{0,k} = 1.$
We implement this as a memoized recurrence in our favorite CAS and
obtain for the pair $(n,k) = (2021, 10)$
$$\bbox[5px,border:2px solid #00A000]{
H_{2021, 10} = 75434038103498084111.}$$
Seeing that we are in the eleventh month of the year $2022$ we also
find
$$\bbox[5px,border:2px solid #00A000]{
H_{2022, 11} = 1212319110727843596031.}$$
Reply to comments, some time later. It was asked why we
differentiate. Well if we have an equation for a generating function it
defines the coefficients being the same on the LHS and the RHS. So that
does not tell us about the relation between the coefficients. If we then
differentiate however we obtain some type of relation between
coefficients that often leads to a recurrence. So that's why. It was
also asked why we extract coefficients. The coefficient gives us
the desired number of sets that sum to at most $n$ so it is the quantity we are interested in. Finally, how do we extract coefficients, we get for example
$$[z^{n-1}] \frac{1}{z} \sum_{q=1}^k \frac{q z^q}{1-z^q} J(z)
= [z^n] \sum_{q=1}^k \frac{q z^q}{1-z^q} J(z)
\\ = [z^n] \sum_{q=1}^k q (z^q + z^{2q} + z^{3q} + \cdots) J(z)
\\ = [z^n] \sum_{q=1}^k q (z^q + z^{2q} + z^{3q} +
\cdots + z^{\lfloor n/q\rfloor q}) J(z)
\\ = \sum_{q=1}^k q
[z^n] \sum_{p=1}^{\lfloor n/q \rfloor} z^{pq} J(z)
\\ = \sum_{q=1}^k q
\sum_{p=1}^{\lfloor n/q \rfloor} [z^{n-pq}] J(z)
= \sum_{q=1}^k q
\sum_{p=1}^{\lfloor n/q \rfloor} J_{n-pq, k}.$$
