Binomial Coefficient is Fun

AtCoder

IDarc110_d

Time2000ms

Memory256MB

Difficulty—

We have a sequence $A$ of $N$ non-negative integers. Compute the sum of $\prod _{i = 1} ^N \dbinom{B_i}{A_i}$ over all sequences $B$ of $N$ non-negative integers whose sum is at most $M$, and print it modulo ($10^9 + 7$). Here, $\dbinom{B_i}{A_i}$, the binomial coefficient, denotes the number of ways to choose $A_i$ objects from $B_i$ objects, and is $0$ when $B_i < A_i$. ## Constraints * All values in input are integers. * $1 \leq N \leq 2000$ * $1 \leq M \leq 10^9$ * $0 \leq A_i \leq 2000$ ## Input Input is given from Standard Input in the following format: $N$ $M$ $A_1$ $A_2$ $\ldots$ $A_N$ [samples]

Samples

Input #1

3 5
1 2 1

Output #1

8

There are four sequences $B$ such that $\prod _{i = 1} ^N \dbinom{B_i}{A_i}$ is at least $1$:

*   $B = {1, 2, 1}$, where $\prod _{i = 1} ^N \dbinom{B_i}{A_i} = \dbinom{1}{1} \times \dbinom{2}{2} \times \dbinom{1}{1} = 1$;
    
*   $B = {2, 2, 1}$, where $\prod _{i = 1} ^N \dbinom{B_i}{A_i} = \dbinom{2}{1} \times \dbinom{2}{2} \times \dbinom{1}{1} = 2$;
    
*   $B = {1, 3, 1}$, where $\prod _{i = 1} ^N \dbinom{B_i}{A_i} = \dbinom{1}{1} \times \dbinom{3}{2} \times \dbinom{1}{1} = 3$;
    
*   $B = {1, 2, 2}$, where $\prod _{i = 1} ^N \dbinom{B_i}{A_i} = \dbinom{1}{1} \times \dbinom{2}{2} \times \dbinom{2}{1} = 2$.
    

The sum of these is $1 + 2 + 3 + 2 = 8$.

Input #2

10 998244353
31 41 59 26 53 58 97 93 23 84

Output #2

642612171

API Response (JSON)

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    "name": "Binomial Coefficient is Fun",
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      "content": "We have a sequence $A$ of $N$ non-negative integers. Compute the sum of $\\prod _{i = 1} ^N \\dbinom{B_i}{A_i}$ over all sequences $B$ of $N$ non-negative integers whose sum is at most $M$, and print it",
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      "statement_type": "Markdown",
      "content": "We have a sequence $A$ of $N$ non-negative integers.\nCompute the sum of $\\prod _{i = 1} ^N \\dbinom{B_i}{A_i}$ over all sequences $B$ of $N$ non-negative integers whose sum is at most $M$, and print it...",
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