Score : $1000$ points

Problem Statement

Given are integers $N$ and $M$. Find the number, modulo $(10^9+7)$, of length-$N$ sequences $(A_1,A_2,\ldots,A_N)$ that consist of non-negative integers and satisfy the following conditions:

• $A_1+A_2+\ldots +A_N = M$;
• For every $i$ ($2 \leq i \leq N-1$), $2 A_i \leq A_{i-1} + A_{i+1}$.

Constraints

• $1 \leq N \leq 10^5$
• $1 \leq M \leq 10^5$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $M$

Output

Print the number, modulo $(10^9+7)$, of sequences that satisfy the conditions.

3 3

7

The following seven sequences satisfy the conditions.

• $0,0,3$
• $0,1,2$
• $1,0,2$
• $1,1,1$
• $2,0,1$
• $2,1,0$
• $3,0,0$

10 100

10804516

10000 100000

694681734