Problem:


Score : 600 points

Problem Statement

There are N slimes standing on a number line.
The i-th slime from the left is at position x_i.

It is guaruanteed that 1 \leq x_1 < x_2 < \ldots < x_N \leq 10^{9}.

Niwango will perform N-1 operations. The i-th operation consists of the following procedures:

Choose an integer k between 1 and N-i (inclusive) with equal probability.

Move the k-th slime from the left, to the position of the neighboring slime to the right.

Fuse the two slimes at the same position into one slime.

Find the total distance traveled by the slimes multiplied by (N-1)! (we can show that this value is an integer), modulo (10^{9}+7). If a slime is born by a fuse and that slime moves, we count it as just one slime.

Constraints

2 \leq N \leq 10^{5}

1 \leq x_1 < x_2 < \ldots < x_N \leq 10^{9}

x_i is an integer.

Subtasks

400 points will be awarded for passing the test cases satisfying N \leq 2000.

Input

Input is given from Standard Input in the following format:

N
x_1 x_2 \ldots x_N

Output

Print the answer.

Sample Input 1

3
1 2 3

Sample Output 1

5

With probability \frac{1}{2}, the leftmost slime is chosen in the first operation, in which case the total distance traveled is 2.

With probability \frac{1}{2}, the middle slime is chosen in the first operation, in which case the total distance traveled is 3.

The answer is the expected total distance traveled, 2.5, multiplied by 2!, which is 5.

Sample Input 2

12
161735902 211047202 430302156 450968417 628894325 707723857 731963982 822804784 880895728 923078537 971407775 982631932

Sample Output 2

750927044

Find the expected value multiplied by (N-1)!, modulo (10^9+7).



Sample input:
3
1 2 3


Sample output:
5

Submission status: Accepted

Solution code:
def main(sample_file = ''):

    """ convenient functions
    # for i, a in enumerate(iterable)
    # q, mod = divmod(a, b)
    # divmod(x, y) returns the tuple (x//y, x%y)
    # Higher-order function: reduce(operator.mul, xyz_count, 1)
    # manage median(s) using two heapq https://atcoder.jp/contests/abc127/tasks/abc127_f
    """

    """convenient decorator
    # @functools.lru_cache():
    # to facilitate use of recursive function
        # ex:
        # from functools import lru_cache
        # import sys
        # sys.setrecursionlimit(10**9)
        # @lru_cache(maxsize=None)
        # def fib(n):
        #     if n < 2:
        #         return n
        #     return fib(n-1) + fib(n-2)
        # print(fib(1000))
    """
    
    # import numpy as np
    import sys
    sys.setrecursionlimit(10**7)
    from itertools import accumulate, combinations, permutations, product # https://docs.python.org/ja/3/library/itertools.html
    # accumulate() returns iterator! to get list: list(accumulate())
    from math import factorial, ceil, floor, sqrt
    def factorize(n):
        """return the factors of the Arg and count of each factor
        
        Args:
            n (long): number to be resolved into factors
        
        Returns:
            list of tuples: factorize(220) returns [(2, 2), (5, 1), (11, 1)]
        """
        fct = []  # prime factor
        b, e = 2, 0  # base, exponent
        while b * b <= n:
            while n % b == 0:
                n = n // b
                e = e + 1
            if e > 0:
                fct.append((b, e))
            b, e = b + 1, 0
        if n > 1:
            fct.append((n, 1))
        return fct
    def combinations_count(n, r):
        """Return the number of selecting r pieces of items from n kinds of items.
        
        Args:
            n (long): number
            r (long): number
        
        Raises:
            Exception: not defined when n or r is negative
        
        Returns:
            long: number
        """
        # TODO: How should I do when n - r is negative?
        if n < 0 or r < 0:
            raise Exception('combinations_count(n, r) not defined when n or r is negative')
        if n - r < r: r = n - r
        if r < 0: return 0
        if r == 0: return 1
        if r == 1: return n
        numerator = [n - r + k + 1 for k in range(r)]
        denominator = [k + 1 for k in range(r)]
        for p in range(2,r+1):
            pivot = denominator[p - 1]
            if pivot > 1:
                offset = (n - r) % p
                for k in range(p-1,r,p):
                    numerator[k - offset] /= pivot
                    denominator[k] /= pivot
        result = 1
        for k in range(r):
            if numerator[k] > 1:
                result *= int(numerator[k])
        return result
    def combinations_with_replacement_count(n, r):
        """Return the number of selecting r pieces of items from n kinds of items allowing individual elements to be repeated more than once.
        
        Args:
            n (long): number
            r (long): number
        
        Raises:
            Exception: not defined when n or r is negative
        
        Returns:
            long: number
        """
        if n < 0 or r < 0:
            raise Exception('combinations_with_replacement_count(n, r) not defined when n or r is negative')
        elif n == 0:
            return 1
        else:
            return combinations_count(n + r - 1, r)
    from bisect import bisect_left, bisect_right
    from collections import deque, Counter, defaultdict # https://docs.python.org/ja/3/library/collections.html#collections.deque
    from heapq import heapify, heappop, heappush, heappushpop, heapreplace,nlargest,nsmallest # https://docs.python.org/ja/3/library/heapq.html
    from copy import deepcopy, copy # https://docs.python.org/ja/3/library/copy.html
    import operator
    from operator import itemgetter #sort 
    # ex1: List.sort(key=itemgetter(1))
    # ex2: sorted(tuples, key=itemgetter(1,2))
    from functools import reduce, lru_cache
    def chmin(x, y):
        """change minimum
        if x > y, x = y and return (x, True).
        convenient when solving problems of dp[i]
        
        Args:
            x (long): current minimum value
            y (long): potential minimum value
        
        Returns:
            (x, bool): (x, True) when updated, else (x, False)
        """
        if x > y:
            x = y
            return (x, True)
        else:
            return (x, False)
    def chmax(x, y):
        """change maximum
        if x < y, x = y and return (x, True).
        convenient when solving problems of dp[i]
        
        Args:
            x (long): current maximum value
            y (long): potential maximum value
        
        Returns:
            (x, bool): (x, True) when updated, else (x, False)
        """
        if x < y:
            x = y
            return (x, True)
        else:
            return (x, False)

    from fractions import gcd # Deprecated since version 3.5: Use math.gcd() instead.
    def gcds(numbers):
        return reduce(gcd, numbers)
    def lcm(x, y):
        return (x * y) // gcd(x, y)
    def lcms(numbers):
        return reduce(lcm, numbers, 1)
    def make_divisors(n, reversed=False):
        """create list of divisors
        
        Args:
            number (int): number from which list of divisors is created
            reversed (bool, optional): ascending order if False. Defaults to False.
        
        Returns:
            list: list of divisors
        """
        divisors = set()
        for i in range(1, int(n**0.5)+1):
            if n % i == 0:
                divisors.add(i)
                divisors.add(n//i)
        return sorted(list(divisors),reverse=reversed)
    
    # first create factorial_list
    # fac_list = mod_factorial_list(n)
    INF = 10 ** 18
    MOD = 10 ** 9 + 7
    modpow = lambda a, n, p = MOD: pow(a, n, p) # Recursive function in python is slow!
    def modinv(a, p = MOD):
        # evaluate reciprocal using Fermat's little theorem:
        # a**(p-1) is identical to 1 (mod p) when a and p is coprime
        return modpow(a, p-2, p)
    def modinv_list(n, p = MOD):
        if n <= 1:
            return [0,1][:n+1]
        else:
            inv_t = [0,1]
            for i in range(2, n+1):
                inv_t += [inv_t[p % i] * (p - int(p / i)) % p]
            return inv_t
    def modfactorial_list(n, p = MOD):
        if n == 0:
            return [1]
        else:
            l = [0] * (n+1)
            tmp = 1
            for i in range(1, n+1):
                tmp = tmp * i % p
                l[i] = tmp
            return l
    def modcomb(n, k, fac_list = [], p = MOD):
        # fac_list = modfactorial_list(100)
        # print(modcomb(100, 5, modfactorial_list(100)))
        from math import factorial
        if n < 0 or k < 0 or n < k: return 0
        if n == 0 or k == 0: return 1
        if len(fac_list) <= n:
            a = factorial(n) % p
            b = factorial(k) % p
            c = factorial(n-k) % p
        else:
            a = fac_list[n]
            b = fac_list[k]
            c = fac_list[n-k]
        return (a * modpow(b, p-2, p) * modpow(c, p-2, p)) % p
    def modadd(a, b, p = MOD):
        return (a + b) % MOD
    def modsub(a, b, p = MOD):
        return (a - b) % p
    def modmul(a, b, p = MOD):
        return ((a % p) * (b % p)) % p
    def moddiv(a, b, p = MOD):
        return modmul(a, modpow(b, p-2, p))

    class UnionFindTree:
        """union find tree class
        TODO: fix this description...
        how to use (example):
        >>  uf = UnionFindTree(N) 
        >>  if uf.find_root(a) == uf.find_root(b):
        >>      do something
        >>  else:
        >>      do something
        >>      uf.unite(a, b)
        """
        def __init__(self, N):
            self.root = [-1] * (N+1)
            self.rank   = [0]  * (N+1)
            self.connected_num = [1] * (N+1)
    
        def find_root(self,x):
            root = self.root
            while root[x] != -1:
                x = root[x]
            return x
    
        def unite(self,x,y):
            root = self.root
            rank = self.rank
            connected_num = self.connected_num
            find_root = self.find_root
    
            rx = find_root(x)
            ry = find_root(y)
            if rx != ry:
                if rank[rx] < rank[ry]:
                    root[rx] = ry
                    rx,ry = ry,rx
                else:
                    if rank[rx] == rank[ry]:
                        rank[rx] += 1
                    root[ry] = rx       
                connected_num[rx] += connected_num[ry]

    # Graph: https://en.wikipedia.org/wiki/Directed_graph
    # Bellman-Ford: O(|V||E|). Use this if there exists an edge with negative length in the graph
    # After N steps, the shortest path has converded if there doesn't exist an cycle of edges with negative
    # Watch out: d[N] == d[2*N] doesn't necessarily mean the graph doesn't have negative cycle
    # ref: https://www.youtube.com/watch?v=1Z6ofKN03_Y

    def BellmanFord(N, M, ABC, vertex_start, vertex_end, value_if_inf = -1, find_shortest = False):
        """to calculate furthest or shortest length between vertex_start and vertex_end using BellmanFord algorithm
        
        Args:
            N (int): number of vertices
            M (int): number of edges
            ABC (list): [(ai, bi, ci) for _ in range(N)] where i-th edge is directed from vertex ai to vertex bi and the length is ci 
            vertex_start (int): start vertex. usually use 0.
            vertex_end (int): end vertex. usually use N-1.
            value_if_inf (int or string as you like, optional): value you want when the furthest (or shortest) distance is infinite (or -infinite). Defaults to -1.
            find_shortest (bool, optional): choose False to find furthest path. Defaults to False.
        
        Returns:
            int or string: normally int (but can be str if you set value_if_inf to str)
            
        Example:
                N, M, P = R()
                ABC = [R() for _ in range(M)]
                ABC = [(a-1, b-1, c-P) for a, b, c in ABC]
                print(BellmanFord(N, M, ABC, 0, N-1, value_if_inf = 'inf'))

        """

        def make_reachable_list(N, M, ABC, vertex_start, vertex_end):
            reachable_to_direct = defaultdict(list)
            reachable_from_direct = defaultdict(list)
            reachable_from_start = [False] * N
            reachable_to_end = [False] * N
            reachable_from_start[vertex_start] = True
            reachable_to_end[vertex_end] = True
            reachable_from_both_sides = [False] * N
            dfs_from_start = []
            dfs_to_end = []
            for a, b, c in ABC:
                reachable_to_direct[a].append(b)
                reachable_from_direct[b].append(a)
                if a == vertex_start:
                    dfs_from_start.append(b)
                    reachable_from_start[b] = True
                if b == vertex_end:
                    dfs_to_end.append(a)
                    reachable_to_end[a] = True
            while dfs_from_start:
                v = dfs_from_start.pop()
                for i in reachable_to_direct[v]:
                    if not reachable_from_start[i]:
                        reachable_from_start[i] = True
                        dfs_from_start.append(i)
            while dfs_to_end:
                v = dfs_to_end.pop()
                for i in reachable_from_direct[v]:
                    if not reachable_to_end[i]:
                        reachable_to_end[i] = True
                        dfs_to_end.append(i)
            for i in range(N):
                reachable_from_both_sides[i] = reachable_from_start[i] and reachable_to_end[i]
            return reachable_from_both_sides
        
        reachable_from_both_sides = make_reachable_list(N, M, ABC, vertex_start, vertex_end)
        
        if find_shortest:
            dist = [INF for i in range(N)]
        else:
            dist = [-INF for i in range(N)]
        
        dist[vertex_start] = 0
        for i in range(N):
            updated = False
            for a, b, c in ABC:
                if not reachable_from_both_sides[a]:
                    continue
                elif find_shortest:
                    update_condition = dist[a] + c < dist[b]
                else:
                    update_condition = dist[a] + c > dist[b]
                if dist[a] != INF and update_condition:
                    dist[b] = dist[a] + c
                    updated = True
                    if i == N-1:
                        return value_if_inf
            if not updated:
                break
        return dist[vertex_end]

    """ initialize variables and set inputs
    # initialize variables
        # to initialize list, use [0] * n
        # to initialize two dimentional array:
            # ex) [[0] * N for _ in range(N)]
            # ex2) dp = [[0] * (N+1) for _ in range(W*2)]
    # set inputs
        # put inputs between specific values (almost as quickly)
        # ex) S = [-INF] + [int(r()) for _ in range(A)] + [INF]
        # open(0).read() is sometimes useful:
        # ex) n, m, *x = map(int, open(0).read().split())
        #     min(x[::2]) - max(x[1::2])
        # ex2) *x, = map(int, open(0).read().split())
        #     don't forget to add comma after *x if only one variable is used
    # preprocessing
        # transpose = [x for x in zip(*data)]
        # ex) [[1, 2, 3], [4, 5, 6], [7, 8, 9]] => [(1, 4, 7), (2, 5, 8), (3, 6, 9)]
        # flat = [flatten for inner in data for flatten in inner]
        # ex) [[1, 2, 3], [4, 5, 6], [7, 8, 9]] => [1, 2, 3, 4, 5, 6, 7, 8, 9]
    # calculate and output
        # output pattern
        # ex1) print(*l) => when l = [2, 5, 6], printed 2 5 6
    """

    # functions to read input
    r = lambda: sys.stdin.readline().strip()
    r_int = lambda: int(r())
    r_float = lambda: float(r())
    R = lambda: list(map(int, r().split()))
    R_str = lambda: r().split()
    R_map = lambda: map(int, r().split())
    R_float = lambda: list(map(float, r().split()))
    R_tuple = lambda: tuple(map(int, r().split()))

    """ how to treat input
    # single int: int(r())
    # single string: r()
    # single float: float(r())
    # line int: R()
    # line string: r().split()
    # line (str, int, int): [j if i == 0 else int(j) for i, j in enumerate(r().split())]
    # lines int: [R() for _ in range(n)]
    """

    # for test
    if sample_file:
        sys.stdin = open(sample_file)
        
    # ----------------------------------
    # main

    N = r_int()
    X = R()
    d = 10**9 + 7
    fac = 1
    for i in range(1, N):
        fac = (fac*i)%d
    l = [fac] * (N)
    for i in range(N-1):
        l[i+1] = (l[i] + moddiv(1,i+2,d)*fac)%d
    ans = 0
    for i in range(N-1):
        ans += (X[i+1]-X[i]) * l[i]
    print(ans%d)

    # to finish code, use return instead of exit()
    # end of main
    # ----------------------------------

    """memo: how to use defaultdict of list
    # initialize
    Dic = defaultdict(list)
    # append / extend
    Dic[x].append(y)
    # three methods for loop: keys(), values(), items()
    for k, v in Dic.items():
    """
    
    """memo: how to solve binary problems
    # to make binary digits text
    >>> a = 5
    >>> bin_str_a = format(a, '#06b')
    >>> print(bin_str_a)
    0b0101 # first 2 strings (='0b') indicates it is binary
    """
    
    """memo: how to solve the problem
    creating simple test/answer
    greed
    simple dp
    graph
    """

if __name__ == '__main__':
    main()

Submission status: Accepted

Solution code:
mod = 10**9+7
rng = 100100
fctr = [1]
finv = [1]
for i in range(1,rng):
  fctr.append(fctr[-1]*i%mod)
for i in range(1,rng):
  finv.append(pow(fctr[i],mod-2,mod))
n = int(input())
a = list(map(int,input().split()))
b = [a[i]-a[i-1] for i in range(1,n)]
lsini = [1,3]
ls = [fctr[n-1],fctr[n-1]*3*finv[2]%mod]
for i in range(2,n-1):
  x = lsini[-1]
  lsini.append(((i+1)*x+fctr[i])%mod)
  ls.append((lsini[-1]*fctr[n-1]*finv[i+1])%mod)
ans = 0
for i in range(n-1):
  ans = (ans + ls[i]*b[i])%mod
print(ans)

Submission status: Accepted

Solution code:
mod = 10**9+7
N = int(input())
x = list(map(int,input().split()))
fact = [1]*(N+1)
for i in range(N):
    fact[i+1] = (i+1)*fact[i]
    fact[i+1] %= mod
inv = [1]*(N+1)
for i in range(N):
    inv[i] = pow(i,mod-2,mod)

x1 = []
for m in range(1,N-1):
    x1.append(x[m]*m*inv[m+1]%mod)
x2 = []
for n in range(N-2):
    x2.append(x[n]*(N-n-2)*inv[N-n-1]%mod)
x3 = []
for n in range(N-1):
    x3.append((x[N-1]-x[n])*inv[N-n-1])
ans = fact[N-1]*(sum(x1)-sum(x2)+sum(x3))
ans %= mod
print(ans)

Submission status: Accepted

Solution code:
mod=10**9+7
n=10**5+100
inv=[0]*(n+1)
inv[1]=1
for i in range(2,n+1):
    inv[i]=-(mod//i)*inv[mod%i]%mod
for i in range(1,n+1):
    inv[i]=(inv[i]+inv[i-1])%mod
N=int(input())
X=[0]+list(map(int,input().split()))
data=[1,1]
for i in range(2,N+1):
    data.append(data[-1]*i%mod)
ans=X[2]-X[1]
for k in range(2,N):
    ans=(k*ans%mod+data[k]*inv[k]*(X[k+1]-X[k])%mod)%mod
print(ans)

Submission status: Accepted

Solution code:
class comb():
  F = [1, 1]
  Fi = [1, 1]
  I = [0, 1]
  def __init__(self, num, mod):
    self.MOD = mod
    for i in range(2, num + 1):
      self.F.append((self.F[-1] * i) % mod)
      self.I.append(mod - self.I[mod % i] * (mod // i) % mod)
      self.Fi.append(self.Fi[-1] * self.I[i] % mod)
  def com(self, n, k):
    if n < k: return 0
    if n < 0 or k < 0: return 0
    return self.F[n] * (self.Fi[k] * self.Fi[n - k] % self.MOD) % self.MOD


N = int(input())
x = list(map(int, input().split()))
MOD = 10 ** 9 + 7


ans = 0
k = 1
for i in range(1, N):
  k = k * i % MOD

com = comb(N, MOD)

t = 0
for i in range(1, N):
  t = (t + k * com.I[i]) % MOD
  ans = (ans + (x[i] - x[i - 1]) * t) % MOD

print(ans)

Submission status: Accepted

Solution code:
N = int(input())
X = list(map(int,input().split()))

mod = 10**9 + 7

n_pow = 1
for i in range(2,N):
    n_pow*=i
    n_pow%=mod

n_s1 = [(n_pow*pow(i,mod-2,mod))%mod for i in range(1,N)]

n_s2 = [n_s1[0]]

s = n_s1[0]

for i in range(1,N-1):
    s+=n_s1[i]
    s%=mod
    n_s2.append(s)

ans = 0

for i in range(N-1):
    ans += (X[i+1]-X[i])*n_s2[i]
    ans %= mod

print(ans)

Submission status: Accepted

Solution code:
import sys
def I(): return int(sys.stdin.readline())
def LI(): return [int(x) for x in sys.stdin.readline().split()]

mod = 10**9+7

def main():
    N = I()
    X = LI()
    D = [X[-1]-x for x in X[:-1]]

    f = 1
    for i in range(1,N):
        f *= i
        f %= mod

    ans = 0
    for i in range(N-1):
        tmp = (D[i] * f)%mod
        tmp *= pow(i+1,mod-2,mod)
        tmp %= mod
        ans += tmp
        ans %= mod
    print(ans)



if __name__ == "__main__":
    main()

Submission status: Accepted

Solution code:
LARGE = 10 ** 9 + 7


def solve(n, x_list):

    n_1_fac = 1
    for i in range(2, n):
        n_1_fac *= i
        n_1_fac %= LARGE

    res = 0
    for i in range(n - 1):
        res += (x_list[n - 1] - x_list[i]) * n_1_fac * pow(n - 1 - i, LARGE - 2, LARGE)
        res %= LARGE

    x_left = 0
    x_right = 0
    for k in range(1, n - 1):
        x_right += x_list[n - 1 - k]
        x_left += x_list[k - 1]
        res += (x_right - x_left) * n_1_fac * pow(k, LARGE - 2, LARGE) * pow(k + 1, LARGE - 2, LARGE)
        res %= LARGE
    return res


def main():
    n = int(input())
    x_list = list(map(int, input().split()))
    res = solve(n, x_list)
    print(res)


def test():
    assert solve(3, [1, 2, 3]) == 5
    assert solve(12, [
        161735902, 211047202, 430302156, 450968417, 628894325, 707723857,
        731963982, 822804784, 880895728, 923078537, 971407775, 982631932
    ]) == 750927044


if __name__ == "__main__":
    test()
    main()

Submission status: Accepted

Solution code:
def main():
    import sys
    input = sys.stdin.readline

    mod = 10**9+7
    # comb init
    nmax = 10 ** 5 + 1  # change here
    fac = [0] * (nmax + 1)
    finv = [0] * (nmax + 1)
    inv = [0] * (nmax + 1)
    fac[0] = 1
    fac[1] = 1
    finv[0] = 1
    finv[1] = 1
    inv[1] = 1
    for i in range(2, nmax):
        fac[i] = fac[i - 1] * i % mod
        inv[i] = mod - inv[mod % i] * (mod // i) % mod
        finv[i] = finv[i - 1] * inv[i] % mod

    N = int(input())
    X = list(map(int, input().split()))

    Y = [0] * (N-1)
    for i in range(N-1):
        Y[i] = X[i+1] - X[i]

    dp = [0] * (N-1)
    cs = [0]
    for i in range(N-1):
        dp[i] = ((cs[i] * inv[i+1])%mod + 1)%mod
        cs.append(cs[-1] + dp[i])
    ans = 0
    for i in range(N-1):
        ans = (ans + ((dp[i] * Y[i])%mod * fac[N-1])%mod)%mod

    print(ans)


if __name__ == '__main__':
    main()

Submission status: Accepted

Solution code:
import sys
input = lambda: sys.stdin.readline().rstrip()
inpl = lambda: list(map(int,input().split()))
def inv_mod(a, p=10**9+7):
    p = abs(p)
    a %= p
    stack = []
    p0 = p
    while a > 1:
        d, a, p = p//a, p%a, a
        stack.append(d)
    x, y = 1, 0
    while stack:
        d = stack.pop()
        x, y = y-d*x, x
    return x % p0
MOD = 10**9 + 7

N = int(input())
x = inpl()
d = [x[i+1]-x[i] for i in range(N-1)]

fac = 1
for i in range(2,N):
    fac = fac * i % MOD
inv = [ inv_mod(i,MOD) for i in range(1,N)]

ans = 0
s = 0
for i in range(N-1):
    s += inv[i]
    ans += d[i]*s
    ans %= MOD
ans = ans * fac % MOD
print(ans)

Submission status: Accepted

Solution code:
import sys
input = sys.stdin.readline

N=int(input())
X=list(map(int,input().split()))

mod=10**9+7

D=[]

for i in range(1,N):
    D.append(X[i]-X[i-1])

FACT=[1]
for i in range(1,2*10**5+1):
    FACT.append(FACT[-1]*i%mod)

INV2=pow(2,mod-2,mod)

K=[1]

for i in range(2,10**5+3):
    K.append(K[-1]+pow(i,mod-2,mod))


ANS=0

for i in range(N-1):
    ANS=(ANS+K[i]*D[i])%mod
    
print(ANS*FACT[N-1]%mod)

Submission status: Accepted

Solution code:
SIZE=100010; MOD=10**9+7 #998244353 #ここを変更する

SIZE += 1
inv = [0]*SIZE  # inv[j] = j^{-1} mod MOD
inv[1] = 1
for i in range(2,SIZE):
    inv[i] = MOD - (MOD//i)*inv[MOD%i]%MOD

# coding: utf-8
# Your code here!
import sys
sys.setrecursionlimit(10**6)
readline = sys.stdin.readline
read = sys.stdin.read

n = int(input())
x = [int(i) for i in readline().split()]

dx =[j-i for i,j in zip(x,x[1:])]

from itertools import accumulate
acc = list(accumulate(inv))

#print(dx)
#print(acc[:10])
ans = 0
for i,di in enumerate(dx):
    ans += di*acc[i+1]
    ans %= MOD

#print(ans)

for i in range(1,n):
    ans *= i
    ans %= MOD
print(ans)

Submission status: Accepted

Solution code:
N = int(input())
X = list(map(int,input().split()))
MOD = 10**9+7

MAXN = N+5
fac = [1,1] + [0]*MAXN
finv = [1,1] + [0]*MAXN
inv = [0,1] + [0]*MAXN
for i in range(2,MAXN+2):
    fac[i] = fac[i-1] * i % MOD
    inv[i] = -inv[MOD%i] * (MOD // i) % MOD
    finv[i] = finv[i-1] * inv[i] % MOD

c = 0
f = fac[N-1]
ans = 0
for i in range(1,N):
    c += f * inv[i]
    c %= MOD
    ans += c * (X[i] - X[i-1])
    ans %= MOD
print(ans)

Submission status: Accepted

Solution code:
nnnnn = 101010
P = 10**9+7

fa = [1]
for i in range(1, nnnnn):
    fa.append(fa[-1] * i % P)
fainv = [pow(fa[-1], P-2, P)]
for i in range(1, nnnnn)[::-1]:
    fainv.append(fainv[-1] * i % P)
fainv = fainv[::-1]

N = int(input())
A = [int(a) for a in input().split()]
B = [A[i+1] - A[i] for i in range(N-1)]
S = [0]
for i in range(1, N):
    S.append((S[-1] + fainv[i] * fa[i-1]) % P)

ans = 0
for i in range(N-1):
    ans = (ans + B[i] * S[i+1]) % P

print(ans * fa[N-1] % P)

Submission status: Accepted

Solution code:
#!usr/bin/env python3
from collections import defaultdict,deque
from heapq import heappush, heappop
from itertools import permutations
import sys
import math
import bisect
import random
def LI(): return [int(x) for x in sys.stdin.readline().split()]
def I(): return int(sys.stdin.readline())
def LS():return [list(x) for x in sys.stdin.readline().split()]
def S():
    res = list(sys.stdin.readline())
    if res[-1] == "\n":
        return res[:-1]
    return res
def IR(n):
    return [I() for i in range(n)]
def LIR(n):
    return [LI() for i in range(n)]
def SR(n):
    return [S() for i in range(n)]
def LSR(n):
    return [LS() for i in range(n)]

sys.setrecursionlimit(1000000)
mod = 1000000007

def solve():
    n = I()
    x = LI()
    fact = [1]
    for i in range(1,n):
        fact.append(fact[-1]*i%mod)
    S = [fact[n-1]*pow(i,mod-2,mod)%mod for i in range(1,n)]
    for i in range(n-2):
        S[i+1] += S[i]
        S[i+1] %= mod
    ans = 0
    for i in range(n-1):
        d = x[i+1]-x[i]
        ans += d*S[i]%mod
        ans %= mod
    print(ans)
    return

#Solve
if __name__ == "__main__":
    solve()

Submission status: Accepted

Solution code:
import sys
readline = sys.stdin.readline

MOD = 10**9+7
def frac(limit):
    frac = [1]*limit
    for i in range(2,limit):
        frac[i] = i * frac[i-1]%MOD
    fraci = [None]*limit
    fraci[-1] = pow(frac[-1], MOD -2, MOD)
    for i in range(-2, -limit-1, -1):
        fraci[i] = fraci[i+1] * (limit + i + 1) % MOD
    return frac, fraci
frac, fraci = frac(1341398)


MOD = 10**9+7
N = int(readline())
X = list(map(int, readline().split()))

D = [x1 - x0 for x0, x1 in zip(X, X[1:])]

g = [0]*(N+3)
g[1] = 1
for i in range(2, N+3):
    g[i] = (g[i-1] + pow(i, MOD-2, MOD)) % MOD

ans = 0
for i in range(N-1):
    ans = (ans + D[i]*g[i+1])%MOD

print(ans*frac[N-1]%MOD)

Submission status: Runtime Error

Solution code:
import numpy as np

stdin = np.array([int(_) for _ in open(0).read().split()])
N, X = stdin[0], stdin[1:]
A = np.array([pow(i, mod - 2, mod) for i in range(N)])
D = X[1:] - X[:-1]
f = 1
for i in range(1, N):
    f *= i
    f %= mod
A = np.cumsum(A)
A *= f
A %= mod
A[1:] *= D
A %= mod
print(np.sum(A) % mod)

Submission status: Runtime Error

Solution code:
import sys
import numpy as np
i8 = np.int64
mod = 10**9 + 7


def main(stdin):
    N, X = stdin[0], stdin[1:]
    D = X[1:] - X[:-1]
    f = 1
    for i in range(1, N):
        f *= i
        f %= mod
    A = np.cumsum(np.array([pow(i, mod - 2, mod) for i in range(N)], dtype=i8), dtype=i8)
    A *= f
    A %= mod
    A[1:] *= D
    A %= mod
    print(np.sum(A) % mod)


if sys.argv[-1] == 'ONLINE_JUDGE':
    import numba
    from numba.pycc import CC
    cc = CC('my_module')
    cc.export('main', '(i8[:],)')(main)
    # b1: bool, i4: int32, i8: int64, double: f8, [:], [:, :]
    cc.compile()
else:
    from my_module import main
    stdin = np.array([int(_) for _ in open(0).read().split()])
    main(stdin)

Submission status: Runtime Error

Solution code:
import sys
import numpy as np
i8 = np.int64
mod = 10**9 + 7


def main(stdin):
    N, X = stdin[0], stdin[1:]
    D = X[1:] - X[:-1]
    f = 1
    for i in range(1, N):
        f *= i
        f %= mod
    A = np.cumsum(np.array([pow(i, mod - 2, mod) for i in range(N)], dtype=i8))
    A *= f
    A %= mod
    A[1:] *= D
    A %= mod
    print(np.sum(A) % mod)


if sys.argv[-1] == 'ONLINE_JUDGE':
    import numba
    from numba.pycc import CC
    cc = CC('my_module')
    cc.export('main', '(i8[:],)')(main)
    # b1: bool, i4: int32, i8: int64, double: f8, [:], [:, :]
    cc.compile()
else:
    from my_module import main
    stdin = np.array([int(_) for _ in open(0).read().split()])
    main(stdin)