Slave to Achievements 3

Kyeonggeun has been playing a web game called isRPG. As the RPG in the name suggests, the game has a lot of equipment items a character can use. Every item is built by gathering materials the way the recipe says, and the options on a finished item are drawn at random from a fixed range. If a finished item is not to his liking, he can dismantle it and get part of the spent materials back.

isRPG has an achievement system, the way most recent games do, so reaching a set number on some counter earns an achievement. The counters Kyeonggeun cares about right now are "craft this many items of equipment" and "dismantle this many items of equipment".

The easiest item to craft in isRPG is the wooden dagger. Its only material is the wood scrap, and the shop sells wood scraps cheaply. The quality of the item has nothing to do with the achievements, so Kyeonggeun decided to carve wooden daggers by the pile.

Spending $N$ wood scraps produces one wooden dagger. Dismantling one wooden dagger returns anywhere from $0$ to $K$ wood scraps. The probability of getting exactly $x$ ($0 \le x \le K$) scraps back does not depend on the value of $x$: it is $1/(K+1)$ for every $x$.

After killing a pile of treasure-hoarding dragons and collecting the loot, Kyeonggeun bought $M$ wood scraps at the shop. He holds no wooden dagger at the moment. From now on he repeats this work: craft wooden daggers from the scraps he holds until he cannot craft a single one more, then dismantle every wooden dagger he holds. Long experience has taught Kyeonggeun that wood scraps sitting in the inventory once all the work is over feel very unpleasant. So he wants to know the probability that 0 wood scraps are left when all the work is over. Having thought that far, he also became curious about the probability that $i$ ($0 \le i < N$) scraps are left. Help him out.

The first line has $N$, $K$ and $M$, separated by spaces. $N$ is the number of wood scraps one wooden dagger costs, and $K$ is the largest number of wood scraps one dismantled wooden dagger returns. ($1 \le K < N \le 10^3$)

$M$ is the number of wood scraps Kyeonggeun starts with. It either satisfies $0 \le M \le 10^{12}$ or equals $-1$. The value $M = -1$ means the answer is the limit each probability converges to as the starting number of wood scraps goes to infinity. The limit exists and is rational, and that has been proven.

On line $i$, print the probability that $i-1$ wood scraps are left. At most $N-1$ scraps are ever left, so the output has exactly $N$ lines.

For an exact check, write the probability as a reduced fraction $\frac{p}{q}$ and print the integer $r$ with $0 \le r < 10^9+7$ that satisfies $p - qr \equiv 0 \pmod{10^9+7}$. Such an integer $r$ exists and is unique in that range, and that has been proven.

In the example with $N = 4$, $K = 2$, $M = 4$, the printed values mean $\frac{1}{3}$, $\frac{1}{3}$, $\frac{1}{3}$, $0$ in that order. In the example with the same $N$ and $K$ and $M = -1$, they mean $\frac{1}{9}$, $\frac{2}{9}$, $\frac{1}{3}$, $\frac{1}{3}$ in that order.