Cat Jumps

AtCoder

IDwtf22_day2_d

Time2000ms

Memory256MB

Difficulty—

You are given a sequence of positive integers $A_1,A_2,\cdots,A_N$. Let $S=N+\sum_{1 \leq i \leq N}A_i$. Cat Snuke has $S$ cards. Each card has an integer written on it. The integers on the cards are $A_1,A_2,\cdots,A_N,-1,\cdots,-1$. Particularly, there are $\sum_{1 \leq i \leq N}A_i$ cards with $-1$. Snuke is now standing at the coordinate $0$ on a number line. He will perform the following operation $S$ times. * Let $x$ be the current coordinate of Snuke. Choose and discard one of the cards he has. Let $v$ be the number on the discarded card, and jump to the coordinate $x+v$. If he has jumped to the coordinate $0$, earn one coin. For each $k=1,2,\cdots,N$, find the number, modulo $998244353$, of sequences of moves for Snuke such that he earns exactly $k$ coins. Note that you should count sequences of moves. Namely, if two cards have the same number, discarding one of them is not distinguished from discarding the other. ## Constraints * $1 \leq N \leq 5000$ * $1 \leq A_i \leq 5000$ * 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

2
1 1

Output #1

2
4

One possible sequence of moves is $(-1,+1,+1,-1)$. Here, Snuke changes his coordinate as $0 \to -1 \to 0 \to 1 \to 0$ and earns two coins.
Here are all possible sequences of moves and the number of coins that will be earned.

*   $(-1,-1,+1,+1)$: one coin
*   $(-1,+1,-1,+1)$: two coins
*   $(-1,+1,+1,-1)$: two coins
*   $(+1,-1,-1,+1)$: two coins
*   $(+1,-1,+1,-1)$: two coins
*   $(+1,+1,-1,-1)$: one coin

Input #2

3
1 2 3

Output #2

140
220
144

Input #3

20
16 6 15 19 1 9 6 1 11 7 6 12 3 11 11 18 10 9 15 5

Output #3

API Response (JSON)

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    "name": "Cat Jumps",
    "description": {
      "content": "You are given a sequence of positive integers $A_1,A_2,\\cdots,A_N$. Let $S=N+\\sum_{1 \\leq i \\leq N}A_i$. Cat Snuke has $S$ cards. Each card has an integer written on it. The integers on the cards are ",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "wtf22_day2_d"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You are given a sequence of positive integers $A_1,A_2,\\cdots,A_N$. Let $S=N+\\sum_{1 \\leq i \\leq N}A_i$.\nCat Snuke has $S$ cards. Each card has an integer written on it. The integers on the cards are ...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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