Score : $900$ points

Problem Statement

How many different sequences of length $2N$, $A = (A_1, A_2, \dots, A_{2N})$, satisfy both of the following conditions?

• The sequence $A$ contains $N$ occurrences of $+1$ and $N$ occurrences of $-1$.
• There are exactly $K$ pairs of $l$ and $r$ $(1 \leq l \leq r \leq 2N)$ such that $A_l + A_{l+1} + \cdots + A_r = 0$.

Constraints

• $1 \leq N \leq 30$
• $1 \leq K \leq N^2$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $K$

Output

Print the number of different sequences that satisfy the conditions in Problem Statement. The answer always fits into the signed $64$-bit integer type.

1 1

2

For $N = 1, K = 1$, two sequences below satisfy the conditions:

• $A = (+1, -1)$
• $A = (-1, +1)$

2 3

2

For $N = 2, K = 3$, two sequences below satisfy the conditions:

• $A = (+1, -1, -1, +1)$
• $A = (-1, +1, +1, -1)$

3 7

6

For $N = 3, K = 7$, six sequences below satisfy the conditions:

• $A = (+1, -1, +1, -1, -1, +1)$
• $A = (+1, -1, -1, +1, +1, -1)$
• $A = (+1, -1, -1, +1, -1, +1)$
• $A = (-1, +1, +1, -1, +1, -1)$
• $A = (-1, +1, +1, -1, -1, +1)$
• $A = (-1, +1, -1, +1, +1, -1)$

8 24

568

30 230

761128315856702

25 455

0

For $N = 25, K = 455$, no sequences satisfy the conditions.