// cs is a list of pairs (c, k) where there's k
// coins of value c.
public static int limitedCoins(int[][] cs, int n) {
    int[] r = new int[n + 1];
    r[0] = 1;
    
    for (int i = 0; i < cs.length; i++) {
        int c = cs[i][0];
        int k = cs[i][1];
        
        int[] rs = r.clone();
        
        for (int j = c; j <= n; j++) {
            rs[j] += rs[j - c];
            // This line effectively performs:
            // r'[j] = sum(r[j - j * c] for j = 0...k)
            // but using rs[] so that the computation is O(1)
            // and in place.
            r[j] += rs[j - c] - (j < c * (k + 1) ? 0 : rs[j - c * (k + 1)]);
        }
    }
    
    return r[n];
}
for (int n = 0; n < 50; n++) {
    System.out.println(n + " " + limitedCoins(new int[][]{{1, 3}, {2, 2}, {5, 3}, {10, 2}}, n));
}

int[] r = new int[n+1];
r[0] = 1;
for (int i = 1; i < r.length; i++) 
    r[i] = 0;