E - Distance on Large Perfect Binary Tree

Score : $500$ points

Problem Statement

We have a tree with $2^N-1$ vertices.
The vertices are numbered $1$ through $2^N-1$. For each $1\leq i < 2^{N-1}$, the following edges exist:

• an undirected edge connecting Vertex $i$ and Vertex $2i$,
• an undirected edge connecting Vertex $i$ and Vertex $2i+1$.

There is no other edge.

Let the distance between two vertices be the number of edges in the simple path connecting those two vertices.

Find the number, modulo $998244353$, of pairs of vertices $(i, j)$ such that the distance between them is $D$.

Constraints

• $2 \leq N \leq 10^6$
• $1 \leq D \leq 2\times 10^6$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $D$

Output

Print the answer.

3 2

14

The following figure describes the given tree.

There are $14$ pairs of vertices such that the distance between them is $2$: $(1,4),(1,5),(1,6),(1,7),(2,3),(3,2),(4,1),(4,5),(5,1),(5,4),(6,1),(6,7),(7,1),(7,6)$.

14142 17320

11284501