Neko Rules the Catniverse (Large Version)

7000ms

262144K

Description:

This problem is same as the previous one, but has larger constraints.

Aki is playing a new video game. In the video game, he will control Neko, the giant cat, to fly between planets in the Catniverse.

There are $$$n$$$ planets in the Catniverse, numbered from $$$1$$$ to $$$n$$$. At the beginning of the game, Aki chooses the planet where Neko is initially located. Then Aki performs $$$k - 1$$$ moves, where in each move Neko is moved from the current planet $$$x$$$ to some other planet $$$y$$$ such that:

Planet $$$y$$$ is not visited yet.
$$$1 \leq y \leq x + m$$$ (where $$$m$$$ is a fixed constant given in the input)

This way, Neko will visit exactly $$$k$$$ different planets. Two ways of visiting planets are called different if there is some index $$$i$$$ such that, the $$$i$$$-th planet visited in the first way is different from the $$$i$$$-th planet visited in the second way.

What is the total number of ways to visit $$$k$$$ planets this way? Since the answer can be quite large, print it modulo $$$10^9 + 7$$$.

Input:

The only line contains three integers $$$n$$$, $$$k$$$ and $$$m$$$ ($$$1 \le n \le 10^9$$$, $$$1 \le k \le \min(n, 12)$$$, $$$1 \le m \le 4$$$) — the number of planets in the Catniverse, the number of planets Neko needs to visit and the said constant $$$m$$$.

Output:

Print exactly one integer — the number of different ways Neko can visit exactly $$$k$$$ planets. Since the answer can be quite large, print it modulo $$$10^9 + 7$$$.

Sample Input:

3 3 1

Sample Output:

Sample Input:

4 2 1

Sample Output:

Sample Input:

5 5 4

Sample Output:

Sample Input:

100 1 2

Sample Output:

Note:

In the first example, there are $$$4$$$ ways Neko can visit all the planets:

$$$1 \rightarrow 2 \rightarrow 3$$$
$$$2 \rightarrow 3 \rightarrow 1$$$
$$$3 \rightarrow 1 \rightarrow 2$$$
$$$3 \rightarrow 2 \rightarrow 1$$$

In the second example, there are $$$9$$$ ways Neko can visit exactly $$$2$$$ planets:

$$$1 \rightarrow 2$$$
$$$2 \rightarrow 1$$$
$$$2 \rightarrow 3$$$
$$$3 \rightarrow 1$$$
$$$3 \rightarrow 2$$$
$$$3 \rightarrow 4$$$
$$$4 \rightarrow 1$$$
$$$4 \rightarrow 2$$$
$$$4 \rightarrow 3$$$

In the third example, with $$$m = 4$$$, Neko can visit all the planets in any order, so there are $$$5! = 120$$$ ways Neko can visit all the planets.

In the fourth example, Neko only visit exactly $$$1$$$ planet (which is also the planet he initially located), and there are $$$100$$$ ways to choose the starting planet for Neko.

Informação

Provedor Codeforces

Código CF1152F2