CF1874E.Jellyfish and Hack

It is well known that quick sort works by randomly selecting a 'pivot' element from the array and partitioning the other elements into two sub-arrays, according to whether they are less than or greater than the pivot. But Jellyfish thinks that choosing a random element is just a waste of time, so she always chooses the first element to be the pivot. The time her code needs to run can be calculated by the following pseudocode:

function fun(A)
if A.length > 0
let L[1 ... L.length] and R[1 ... R.length] be new arrays
L.length = R.length = 0
for i = 2 to A.length
if A[i] < A[1]
L.length = L.length + 1
L[L.length] = A[i]
else
R.length = R.length + 1
R[R.length] = A[i]
return A.length + fun(L) + fun(R)
else
return 0

Now you want to show her that her code is slow. When the function $\mathrm{fun(A)}$ is greater than or equal to $lim$, her code will get $\text{Time Limit Exceeded}$. You want to know how many distinct permutations $P$ of $[1, 2, \dots, n]$ satisfies $\mathrm{fun(P)} \geq lim$. Because the answer may be large, you will only need to find the answer modulo $10^9+7$.

众所周知，快速排序通过从数组中随机选择一个“基准”（pivot）元素，并根据其余元素是否小于或大于该基准，将它们划分为两个子数组。但水母（Jellyfish）认为随机选取元素纯粹是浪费时间，因此她总是选择第一个元素作为基准。她的代码运行所需的时间可通过以下伪代码计算：

function fun(A)
if A.length > 0
let L[1 ... L.length] and R[1 ... R.length] be new arrays
L.length = R.length = 0
for i = 2 to A.length
if A[i] < A[1]
L.length = L.length + 1
L[L.length] = A[i]
else
R.length = R.length + 1
R[R.length] = A[i]
return A.length + fun(L) + fun(R)
else
return 0

现在你想向她证明她的代码很慢。当函数 $\mathrm{fun(A)}$ 的值大于等于 $lim$ 时，她的代码将出现“超时”（Time Limit Exceeded）。你想知道：有多少个 $[1, 2, \dots, n]$ 的不同排列 $P$ 满足 $\mathrm{fun(P)} \geq lim$？由于答案可能很大，你只需输出结果对 $10^9+7$ 取模的值。

The only line of the input contains two integers $n$ and $lim$ ($1 \leq n \leq 200$, $1 \leq lim \leq 10^9$).

输入仅包含一行，其中有两个整数 $n$ 和 $lim$（$1 \leq n \leq 200$，$1 \leq lim \leq 10^9$）。

Output the number of different permutations that satisfy the condition modulo $10^9+7$.

输出满足条件的不同排列的数量，对 $10^9+7$ 取模。

In the first example, $P = [1, 4, 2, 3]$ satisfies the condition, because: $\mathrm{fun([1, 4, 2, 3]) = 4 + fun([4, 2, 3]) = 7 + fun([2, 3]) = 9 + fun([3]) = 10}$

Do remember to output the answer modulo $10^9+7$.

在第一个例子中，$P = [1, 4, 2, 3]$ 满足条件，因为：$\mathrm{fun([1, 4, 2, 3]) = 4 + fun([4, 2, 3]) = 7 + fun([2, 3]) = 9 + fun([3]) = 10}$

请注意，输出答案时需对 $10^9+7$ 取模。