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).

Solution in C (Clang 3.8.0) (judge verdict: Accepted)

#define _CRT_SECURE_NO_WARNINGS

#include <stdbool.h>
#include <stdio.h>
#include <stdlib.h>
#include <inttypes.h>
#include <limits.h>
#include <ctype.h>
#include <stdint.h>
#include <string.h>
#include <wchar.h>
#include <math.h>

#define N_MAX (100)
#define P_MAX (100)
#define DP_ARRAY_SIZE (N_MAX * P_MAX / 32 + 1)

#define MIN(a, b) ((a) < (b) ? (a) : (b))
#define MAX(a, b) ((a) > (b) ? (a) : (b))
#define ABS(a) ((a) < 0 ? -(a) : (a))
#define ABSS(a, b) ((a) > (b) ? (a) - (b) : (b) - (a))
#define SWAP(type, a, b) { const type tmp = a; a = b; b = tmp; }

int compare_sz_asc(const void* a, const void* b) {
    return *((size_t*)a) < *((size_t*)b) ? -1 : 1;
}

int compare_sz_desc(const void* a, const void* b) {
    return *((size_t*)a) > * ((size_t*)b) ? -1 : 1;
}

int compare_i64_asc(const void* a, const void* b) {
    return *((int64_t*)a) < *((int64_t*)b) ? -1 : 1;
}

int compare_i64_desc(const void* a, const void* b) {
    return *((int64_t*)a) > * ((int64_t*)b) ? -1 : 1;
}

int compare_u64_asc(const void* a, const void* b) {
    return *((uint64_t*)a) < *((uint64_t*)b) ? -1 : 1;
}

int compare_u64_desc(const void* a, const void* b) {
    return *((uint64_t*)a) > * ((uint64_t*)b) ? -1 : 1;
}

int compare_c_asc(const void* a, const void* b) {
    return *((char*)a) < *((char*)b) ? -1 : 1;
}

int compare_c_desc(const void* a, const void* b) {
    return *((char*)a) > * ((char*)b) ? -1 : 1;
}

int compare_dp_asc(const void* a, const void* b) {
    return *((double*)a) < *((double*)b) ? -1 : 1;
}

static uint64_t modinvU64(const uint64_t a, const uint64_t m) {
    int64_t aa = (int64_t)a;
    const int64_t mm = (int64_t)m;
    int64_t b = mm;
    int64_t u = 1;
    int64_t v = 0;

    while (b != 0) {
        const int64_t t = aa / b;

        aa -= t * b;
        SWAP(uint64_t, aa, b);
        u -= t * v;
        SWAP(uint64_t, u, v);
    }

    u %= mm;
    if (u < 0) {
        u += m;
    }

    return (uint64_t)u;
}

static size_t powSz(const size_t base, const size_t exp) {
    if (exp == 0) {
        return 1;
    }

    if (exp == 1) {
        return base;
    }

    if (exp % 2 == 0) {
        return powSz(base * base, exp / 2);
    }
    else {
        return base * powSz(base, exp - 1);
    }
}

static uint64_t powU64(const uint64_t base, const uint64_t exp) {
    if (exp == 0) {
        return 1;
    }

    if (exp == 1) {
        return base;
    }

    if (exp % 2 == 0) {
        return powU64(base * base, exp / 2);
    }
    else {
        return base * powU64(base, exp - 1);
    }
}

static uint64_t powU64WMod(const uint64_t base, const uint64_t exp, const uint64_t m) {
    if (exp == 0) {
        return 1;
    }

    if (exp == 1) {
        return base % m;
    }

    if (exp % 2 == 0) {
        return powU64WMod(base * base % m, exp / 2, m);
    }
    else {
        return base * powU64WMod(base, exp - 1, m) % m;
    }
}

static size_t comb(const size_t n, const size_t r) {
    size_t result = 1;

    for (size_t i = 0; i < r; i++) {
        result *= n - i;
        result /= i + 1;
    }

    return result;
}

static uint64_t combU64(const uint64_t n, const uint64_t r) {
    uint64_t result = 1;

    for (uint64_t i = 0; i < r; i++) {
        result *= n - i;
        result /= i + 1;
    }

    return result;
}

static uint64_t combU64WMod(const uint64_t n, const uint64_t r, const uint64_t m) {
    uint64_t result = 1;
    uint64_t inv = 1;

    for (uint64_t i = 0; i < r; i++) {
        result = result * (n - i) % m;
        inv = inv * (i + 1) % m;
    }

    return result * modinvU64(inv, m) % m;
}

static uint64_t permU64WMod(const uint64_t n, const uint64_t m) {
    uint64_t result = 1;

    for (uint64_t v = 2; v <= n; v++) {
        result *= v;
        result %= m;
    }

    return result;
}

static size_t gcdZu(size_t m, size_t n) {
    size_t temp;

    while (m % n != 0) {
        temp = n;
        n = m % n;
        m = temp;
    }

    return n;
}

static uint64_t gcdU64(uint64_t m, uint64_t n)
{
    uint64_t temp;

    while (m % n != 0) {
        temp = n;
        n = m % n;
        m = temp;
    }

    return n;
}

static uint64_t popcntU64(uint64_t val) {
    uint64_t cnt = 0;

    while (val != 0) {
        if ((val & UINT64_C(0x1)) != 0) {
            cnt++;
        }

        val >>= UINT64_C(0x1);
    }

    return cnt;
}

typedef struct {
    uint64_t val;
}BHeap_Val;

#define BHEAP_VAL_TYPE BHeap_Val

#define BHEAP_INVALID_NODE ((size_t)100000000)

typedef struct {
    BHEAP_VAL_TYPE val;
} BHeap_Node;


struct _BHeap_Ctx {
    size_t size;
    int (*compare)(const BHEAP_VAL_TYPE* parent, const BHEAP_VAL_TYPE* node);
    void(*notifyNodeChange)(const struct _BHeap_Ctx* const ctx, const size_t nodeListSize, const size_t* const nodeList);
    BHeap_Node* node;
};

typedef struct _BHeap_Ctx BHeap_Ctx;

static void bHeap_Init(BHeap_Ctx* const ctx, int(*compare)(const BHEAP_VAL_TYPE* parent, const BHEAP_VAL_TYPE* node), void(*notifyNodeChange)(const BHeap_Ctx* const ctx, const size_t nodeListSize, const size_t* const nodeList), BHeap_Node* const nodes) {
    ctx->size = 0;
    ctx->compare = compare;
    ctx->notifyNodeChange = notifyNodeChange;
    ctx->node = nodes;
}

static void bHeap_Swap(BHeap_Ctx* const ctx, const size_t node0, const size_t node1) {
    const BHEAP_VAL_TYPE tmp = ctx->node[node0].val;
    ctx->node[node0].val = ctx->node[node1].val;
    ctx->node[node1].val = tmp;

    if (ctx->notifyNodeChange != NULL) {
        const size_t nodeList[2] = { node0,node1 };
        ctx->notifyNodeChange(ctx, 2, nodeList);
    }
}

static void bHeap_UpHeap(BHeap_Ctx* const ctx, const size_t nodeIdx) {
    if (nodeIdx == 0) {
        return;
    }

    const size_t parentIdx = (nodeIdx + 1) / 2 - 1;

    const int compareResult = ctx->compare(&(ctx->node[parentIdx].val), &(ctx->node[nodeIdx].val));

    if (0 < compareResult) {
        bHeap_Swap(ctx, nodeIdx, parentIdx);

        if (parentIdx != 0) {
            bHeap_UpHeap(ctx, parentIdx);
        }
    }
}

static void bHeap_DownHeap(BHeap_Ctx* const ctx, const size_t nodeIdx) {
    if (ctx->size - 1 <= nodeIdx) {
        return;
    }

    const size_t leftIdx = 2 * (nodeIdx + 1) - 1;

    if (leftIdx < ctx->size) {
        const int leftResult = ctx->compare(&(ctx->node[nodeIdx].val), &(ctx->node[leftIdx].val));

        size_t swapTarget = leftResult < 0 ? BHEAP_INVALID_NODE : leftIdx;

        const size_t rightIdx = leftIdx + 1;

        if (rightIdx < ctx->size) {
            const int rightResult = ctx->compare(&(ctx->node[nodeIdx].val), &(ctx->node[rightIdx].val));

            if (swapTarget == BHEAP_INVALID_NODE) {
                if (0 < rightResult) {
                    swapTarget = rightIdx;
                }
            }
            else {
                if (0 < rightResult) {
                    const int rightLeftResult = ctx->compare(&(ctx->node[leftIdx].val), &(ctx->node[rightIdx].val));

                    if (0 < rightLeftResult) {
                        swapTarget = rightIdx;
                    }
                }
            }
        }

        if (swapTarget != BHEAP_INVALID_NODE) {
            bHeap_Swap(ctx, nodeIdx, swapTarget);

            bHeap_DownHeap(ctx, swapTarget);
        }
    }
}

static void bHeap_Add(BHeap_Ctx* const ctx, const BHEAP_VAL_TYPE val) {
    ctx->node[ctx->size].val = val;

    ctx->size++;

    if (ctx->notifyNodeChange != NULL) {
        const size_t node = ctx->size - 1;
        ctx->notifyNodeChange(ctx, 1, &node);
    }

    bHeap_UpHeap(ctx, ctx->size - 1);
}

static void bHeap_RemoveTop(BHeap_Ctx* const ctx) {
    if (ctx->size == 0) {
        return;
    }

    if (ctx->size == 1) {
        ctx->size--;
        return;
    }

    ctx->node[0].val = ctx->node[ctx->size - 1].val;

    if (ctx->notifyNodeChange != NULL) {
        const size_t node = 0;
        ctx->notifyNodeChange(ctx, 1, &node);
    }

    ctx->size--;

    bHeap_DownHeap(ctx, 0);
}

static void bHeap_RemoveNode(BHeap_Ctx* const ctx, const size_t node) {
    if (node == ctx->size - 1) {
        ctx->size--;
    }
    else {
        ctx->node[node].val = ctx->node[ctx->size - 1].val;

        ctx->size--;

        if (ctx->notifyNodeChange != NULL) {
            ctx->notifyNodeChange(ctx, 1, &node);
        }

        bHeap_DownHeap(ctx, node);
        bHeap_UpHeap(ctx, node);
    }
}

static BHEAP_VAL_TYPE* bHeap_GetTopVal(const BHeap_Ctx* const ctx) {
    return ctx->size == 0 ? NULL : &(ctx->node[0].val);
}

typedef struct {
    uint64_t cnt;
    size_t* child;
    size_t numChild;
    size_t maxNumChild;
} Tree_Node;

#define TREE_INIT_ALLOC (128)

static Tree_Node tree_Node[200000];

static void tree_Init(const uint64_t N) {
    for (uint64_t i = 0; i < N; i++) {
        tree_Node[i].cnt = 0;
        tree_Node[i].numChild = 0;
        tree_Node[i].child = malloc(sizeof(size_t) * TREE_INIT_ALLOC);
        tree_Node[i].maxNumChild = TREE_INIT_ALLOC;
    }
}

static void tree_AddChild(const size_t parent, const size_t child) {
    if (tree_Node[parent].maxNumChild <= tree_Node[parent].numChild) {
        tree_Node[parent].child = realloc(tree_Node[parent].child, sizeof(size_t) * tree_Node[parent].maxNumChild * 2);
        tree_Node[parent].maxNumChild *= 2;
    }

    tree_Node[parent].child[tree_Node[parent].numChild] = child;
    tree_Node[parent].numChild++;
}

static BHeap_Node BN_Asc[200000];
static BHeap_Node BN_Desc[200000];

static int _bHeap_Compare_Asc(const BHEAP_VAL_TYPE* parent, const BHEAP_VAL_TYPE* node) {
    return parent->val <= node->val ? -1 : 1;
}

static int _bHeap_Compare_Desc(const BHEAP_VAL_TYPE* parent, const BHEAP_VAL_TYPE* node) {
    return parent->val >= node->val ? -1 : 1;
}

static size_t PBHeap_Asc_Idx[200000];
static size_t PBHeap_Desc_Idx[200000];

static void _bHeap_NotifyNodeChange_Asc(const BHeap_Ctx* const ctx, const size_t nodeListSize, const size_t* const nodeList) {
    for (size_t i = 0; i < nodeListSize; i++) {
        PBHeap_Asc_Idx[ctx->node[nodeList[i]].val.val] = nodeList[i];
    }
}

static void _bHeap_NotifyNodeChange_Desc(const BHeap_Ctx* const ctx, const size_t nodeListSize, const size_t* const nodeList) {
    for (size_t i = 0; i < nodeListSize; i++) {
        PBHeap_Desc_Idx[ctx->node[nodeList[i]].val.val] = nodeList[i];
    }
}

#define MOD_VAL (UINT64_C(1000000007))

#define UF_ROOT ((size_t)10000000)

typedef struct {
    size_t parent;
    uint64_t w;
    uint64_t v;
    uint64_t wSum;
    uint64_t vSum;
} UF_Node;

static UF_Node uf_Node[101];

static size_t uf_GetRoot(const size_t idx) {
    if (uf_Node[idx].parent == UF_ROOT) {
        return idx;
    }

    const size_t root = uf_GetRoot(uf_Node[idx].parent);

    uf_Node[idx].parent = root;

    return root;
}

static void uf_Merge(const size_t a, const size_t b) {
    const size_t aRoot = uf_GetRoot(a);
    const size_t bRoot = uf_GetRoot(b);

    if (aRoot != bRoot) {
        uf_Node[aRoot].parent = bRoot;

        uf_Node[bRoot].wSum += uf_Node[aRoot].wSum;
        uf_Node[bRoot].vSum += uf_Node[aRoot].vSum;
    }
}

#define MOD_VAL (UINT64_C(1000000007))

static uint64_t x[100000];
static uint64_t probTable[100001];

int main(void) {
    uint64_t N;

    scanf("%"PRIu64, &N);
    for (uint64_t i = 0; i < N; i++) {
        scanf("%"PRIu64, &x[i]);
    }

    const uint64_t N1 = permU64WMod(N - 1, MOD_VAL);

    for (uint64_t i = 0; i < N; i++) {
        if (i == 0) {
            probTable[i] = N1;
        }
        else {
            probTable[i] = (probTable[i - 1] + N1 * modinvU64(i + 1, MOD_VAL) % MOD_VAL) % MOD_VAL;
        }
    }

    uint64_t res = 0;

    for (uint64_t i = 0; i < N - 1; i++) {
        res = (res + (x[i + 1] - x[i]) * probTable[i]) % MOD_VAL;
    }

    printf("%"PRIu64"\n", res);

    return 0;
}