Subsequence and Prefix Sum

AtCoder

IDarc180_c

Time2000ms

Memory256MB

Difficulty—

You are given an integer sequence $A=(A_1,A_2,\cdots,A_N)$ of length $N$. You will perform the following operation exactly once: * Choose a non-empty subsequence of $A$ (not necessarily contiguous) and replace it with its cumulative sums. More precisely, first choose a sequence of indices $(i_1,i_2,\cdots,i_k)$ such that $1 \leq i_1 < i_2 < \cdots < i_k \leq N$. The length of the sequence $k$ ($1 \leq k \leq N$) can be chosen freely. Then, for each $j$ ($1 \leq j \leq k$), replace the value of $A_{i_j}$ with $\sum_{1 \leq x \leq j} A_{i_x}$. This replacement is done simultaneously for all chosen indices. Find, modulo $10^9+7$, the number of possible sequences $A$ after the operation. ## Constraints * $1 \leq N \leq 100$ * $-10 \leq A_i \leq 10$ * All input values are integers. ## Input The input is given from Standard Input in the following format: $N$ $A_1$ $A_2$ $\cdots$ $A_N$ [samples]

Samples

Input #1

3
1 1 2

Output #1

4

The possible sequences $A$ after the operation are as follows:

*   $A=(1,1,2)$: This can be achieved with $k=1$ and $(i_1)=(1)$.
*   $A=(1,2,2)$: This can be achieved with $k=2$ and $(i_1,i_2)=(1,2)$.
*   $A=(1,1,3)$: This can be achieved with $k=2$ and $(i_1,i_2)=(1,3)$.
*   $A=(1,2,4)$: This can be achieved with $k=3$ and $(i_1,i_2,i_3)=(1,2,3)$.

Input #2

4
1 -1 1 -1

Output #2

Input #3

5
0 0 0 0 0

Output #3

Input #4

40
2 -2 1 3 -3 -1 -2 -3 0 -1 -2 0 -3 0 0 2 0 -1 2 -2 -2 -1 3 -2 -2 -2 2 3 2 -3 0 -2 2 1 3 0 -1 0 -2 -3

Output #4

420429545

API Response (JSON)

{
  "problem": {
    "name": "Subsequence and Prefix Sum",
    "description": {
      "content": "You are given an integer sequence $A=(A_1,A_2,\\cdots,A_N)$ of length $N$. You will perform the following operation exactly once: *   Choose a non-empty subsequence of $A$ (not necessarily contiguous)",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc180_c"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You are given an integer sequence $A=(A_1,A_2,\\cdots,A_N)$ of length $N$.\nYou will perform the following operation exactly once:\n\n*   Choose a non-empty subsequence of $A$ (not necessarily contiguous)...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments