Q:

Summing up Problem (Combinations)

This is a part of a bigger problem I was solving.
Problem: $N$ is a positive integer. There are $k$ number of other positive integers ($\le N$)
In how many ways can you make $N$ by summing up any number of those $k$ integers. You can use any integer, any number of times.
For example: $N = 10$, $k=1: \{ 1 \}$
then there's only $1$ way of making $10$ using integers in braces: $1+1+1+1+\cdots+1 = 10$
another example: $N = 10$, $k = 2: \{ 1, 3\}$
number of ways $= 4$:
$1,1,1,1,1,1,1,1,1,1$
$1,1,1,1,1,1,1,3$
$1,1,1,1,3,3$
$1,3,3,3$
The question is to derive a generalized logic/formula to calculate the number of ways.

A:

There is a simple recursive formula for that problem.
$F(0, k) = 1$
$F(N, \emptyset) = 0$
$F(N, k) = F(N - \min(k), k) + F(N, k\backslash \{\min(k)\})$

