Problem:


Score : 100 points

Problem Statement

There are N children, numbered 1, 2, \ldots, N.

They have decided to share K candies among themselves.
Here, for each i (1 \leq i \leq N), Child i must receive between 0 and a_i candies (inclusive).
Also, no candies should be left over.

Find the number of ways for them to share candies, modulo 10^9 + 7.
Here, two ways are said to be different when there exists a child who receives a different number of candies.

Constraints

All values in input are integers.

1 \leq N \leq 100

0 \leq K \leq 10^5

0 \leq a_i \leq K

Input

Input is given from Standard Input in the following format:

N K
a_1 a_2 \ldots a_N

Output

Print the number of ways for the children to share candies, modulo 10^9 + 7.

Sample Input 1

3 4
1 2 3

Sample Output 1

5

There are five ways for the children to share candies, as follows:

(0, 1, 3)

(0, 2, 2)

(1, 0, 3)

(1, 1, 2)

(1, 2, 1)

Here, in each sequence, the i-th element represents the number of candies that Child i receives.

Sample Input 2

1 10
9

Sample Output 2

0

There may be no ways for the children to share candies.

Sample Input 3

2 0
0 0

Sample Output 3

1

There is one way for the children to share candies, as follows:

(0, 0)

Sample Input 4

4 100000
100000 100000 100000 100000

Sample Output 4

665683269

Be sure to print the answer modulo 10^9 + 7.



Sample input:
3 4
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_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, K = R()
    A = R()
    
    dp = [[0]*(N+1) for _ in range(K+1)]
    for i in range(N+1):
        dp[0][i] = 1
    for j, a in enumerate(A):
        for i in range(K):
            if (i)-a >= 0:
                dp[i+1][j+1] = (dp[i][j+1] + dp[i+1][j] - dp[i-a][j]) % (10**9+7)
            else:
                dp[i+1][j+1] = (dp[i][j+1] + dp[i+1][j]) % (10**9+7)
    print(dp[K][N])

    # 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:
N, K = map(int, input().split())
a = list(map(int, input().split()))
mod = 10**9+7

dp = [[0 for _ in range(K+1)] for _ in range(N+1)]
dp[0][0] = 1
dp2 = [[0 for _ in range(K+2)] for _ in range(N)]
for i in range(N):
    for j in range(K+1):
        dp2[i][j+1] = (dp2[i][j] + dp[i][j]) % mod
    for j in range(K+1):
        dp[i+1][j] = (dp[i+1][j] + (dp2[i][j+1] - dp2[i][max(j - a[i], 0)])%mod) % mod
print(dp[-1][-1])

Submission status: Accepted

Solution code:
N,K = map(int, input().split())
A = [int(a) for a in input().split()]

mod = 10**9+7

dp = [[0]*(K+1) for _ in range(N+1)]
dp[0][0] = 1
    
for i in range(N):
    S = [0]*(K+2)
    for j in range(K+1):
        S[j+1] = S[j] + dp[i][j]
        S[j+1] %= mod
    for j in range(K+1):
        dp[i+1][j] = S[j+1]-S[max(j-A[i], 0)]
        dp[i+1][j] %= mod
        
print(dp[N][K])

Submission status: Accepted

Solution code:
import sys
input = sys.stdin.readline
n,k=[int(j) for j in input().split()]
a=[int(j) for j in input().split()]
mod=10**9+7
import numpy as np
dp=np.zeros((n+1,k+1),dtype=np.int64)
dp[0][0]=1
for i in range(n):
    dp[i+1]=dp[i]
    dp[i+1,a[i]+1:]-=dp[i,:-(a[i]+1)]
    np.cumsum(dp[i+1],out=dp[i+1])
    dp[i+1]%=mod
print(dp[n][k])

Submission status: Accepted

Solution code:
p = 10**9+7
N,K = map(int,input().split())
A = list(map(int,input().split()))
A.insert(0,0)
dp = [[0 for _ in range(K+1)] for _ in range(N+1)]
dq = [[0 for _ in range(K+2)] for _ in range(N+1)]
for j in range(K+1):
    if j>A[N]:
        dp[N][j] = 0
    else:
        dp[N][j] = 1
    dq[N][j+1] = (dq[N][j]+dp[N][j])%p
for i in range(N-1,0,-1):
    for j in range(K+1):
        dp[i][j] = (dq[i+1][j+1]-dq[i+1][max(0,j-A[i])])%p
        dq[i][j+1] = (dq[i][j]+dp[i][j])%p
print(dp[1][K])

Submission status: Accepted

Solution code:
MOD = 10**9+7
n, k = map(int, input().split())
a = list(map(int, input().split()))
dp = [0] * (k + 1)
# dp[i] - the number of ways such that we used i candies so far
dp[0] = 1
for i in range(n):
    fake = [0] * (k + 1)
    for used in range(k, -1, -1):
        tmp = dp[used]
        L = used + 1
        R = used + min(a[i], k - used)
        if L <= R:
            fake[L] = (fake[L] + tmp) % MOD
            if R + 1 <= k:
                fake[R + 1] = (fake[R + 1] - tmp + MOD) % MOD
    prefix_sum = 0
    for j in range(k + 1):
        prefix_sum = (prefix_sum + fake[j]) % MOD
        dp[j] = (dp[j] + prefix_sum) % MOD
print(dp[k])

Submission status: Accepted

Solution code:
def p_m():
    n, k = map(int, input().split())
    *a, = map(int, input().split())
    mod = 10 ** 9 + 7
    # dp[i][j]はi番目の子にj個の飴を配る方法が何通りかどうか
    dp = [[0] * (k + 1) for _ in range(n + 1)]
    for i in range(n + 1):
        dp[i][0] = 1

    for i in range(1, n + 1):
        for j in range(1, k + 1):
            if j - 1 - a[i - 1] >= 0:
                dp[i][j] = (dp[i - 1][j] + dp[i][j - 1] - dp[i - 1][j - 1 - a[i - 1]]) % mod
            else:
                dp[i][j] = (dp[i - 1][j] + dp[i][j - 1]) % mod
    print(dp[n][k])


if __name__ == '__main__':
    p_m()

Submission status: Accepted

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

n,k=map(int, input().split()) 
a=list(map(int, input().split()))
mod=10**9+7
dp=[[0]*(k+1) for i in range(n+1)]

for i in range(n+1):
    dp[i][0]=1



for i in range(1,n+1):
    for j in range(1,k+1):
        if j-1-a[i-1]>=0:
            dp[i][j]=(dp[i][j-1]+dp[i-1][j]-dp[i-1][j-1-a[i-1]])%mod
        
        else:
            dp[i][j]=(dp[i][j-1]+dp[i-1][j])%mod

print(dp[-1][-1]%mod)

Submission status: Accepted

Solution code:
def cumsum(inlist):
    s = 0
    outlist = [0]
    for i in inlist:
        s += i
        outlist.append(s)
    return outlist

mod = 10**9+7
n, k = [ int(v) for v in input().split() ]
a_list = [ int(v) for v in input().split() ]

dp_list = [ 0 for i in range(k+1) ]
dp_list[0] = 1

for i in range(n):
    cum_dp_list = cumsum(dp_list)

    new_dp_list = [ 0 for i in range(k+1) ]
    for j in range(k+1):
        r = min(j+1,k+1)
        l = max(0,j-a_list[i]) 
        new_dp_list[j] = cum_dp_list[r] - cum_dp_list[l]
    dp_list = [ i % mod for i in new_dp_list ]

print(dp_list[-1])

Submission status: Accepted

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

MOD = int(pow(10, 9) + 7)

dp = [[0 for i in range(K+1)] for i in range(N+1)]
dp[0] = [1 for i in range(K+1)]

for c in range(1, N+1):
    for r in range(K, -1, -1):
        lo = dp[c - 1][r]
        hi = dp[c - 1][r + candies[c - 1] + 1] if r + candies[c - 1] + 1 <= K else 0
        dp[c][r] = (lo - hi) + (dp[c][r+1] if r + 1 <= K else 0)
        dp[c][r] %= MOD

def pprint(table):
    for i, row in enumerate(table):
        print([row[j] - row[j+1] for j in range(K)] + [row[K]])

# pprint(dp)

print((dp[N][0] - (dp[N][1] if K > 0 else 0)) % MOD)

Submission status: Accepted

Solution code:
mod = 10**9+7
N, K = map(int, input().split())
a = list(map(int, input().split()))
dp = [0]*(K+1)

for i in range(a[0]+1):
    dp[i] = 1
for i in range(1, N):
    S = [0]*(K+2)
    for j in range(1, K+2):
        S[j] = S[j-1]+dp[j-1]
        S[j]%=mod
    for j in range(K+1):
        dp[j] = (S[j+1]-S[max(0, j-a[i])])%mod
print(dp[-1])

Submission status: Accepted

Solution code:
def main():
    N, K = map(int, input().split())
    A = map(int, input().split())
    u = [1] + [0] * K
    v = [1] + [0] * K
    s = 1
    for a in A:
        u, v = v, u
        s += a
        c = 0
        for i, t in enumerate(u[:a + 1]):
            c += t
            v[i] = c
        for i, t, tt in zip(range(a + 1, K + 1), u[a + 1:s], u):
            c += t - tt
            v[i] = c
    print(v[K] % (10 ** 9 + 7))

main()

Submission status: Accepted

Solution code:
def main():
    N, K = map(int, input().split())
    a = tuple(map(int, input().split()))
    l = [1 if a[0] >= j else 0 for j in range(K+1)]
    lt= [1 if a[0] >= j else 0 for j in range(K+1)]
    s = a[0]
    for i in a[1:]:
        lt, l = l, lt
        s += i
        l[0] = 1
        t = 1
        for j in range(1, i+1):
                t += lt[j]
                l[j] = t
        for j in range(i+1, min(K+1, s+1)):
                t += (lt[j] - lt[j - i - 1])
                l[j] = t
    return l[-1] % (10 ** 9 + 7)
            
print(main())

Submission status: Accepted

Solution code:
N, K = map(int, input().split())
a = tuple(map(int, input().split()))

def main():
    l = [1 if a[0] >= j else 0 for j in range(K+1)]
    lt= [0] * (K+1)
    s = a[0]
    for i in range(1,N):
        lt, l = l, lt
        s += a[i]
        l[0] = 1
        t = 1
        for j in range(1, a[i]+1):
                t += lt[j]
                l[j] = t
        for j in range(a[i]+1, min(K+1, s+1)):
                t += (lt[j] - lt[j - a[i] - 1])
                l[j] = t
    return l[-1] % (10 ** 9 + 7)
            
print(main())

Submission status: Accepted

Solution code:
def main():
    N, K = map(int, input().split())
    A = list(map(int, input().split()))
    u = [1] + [0] * K
    v = [1] + [0] * K
    s = 0
    for i in A:
        u, v = v, u
        s += i
        t = 0
        for k in range(i):
            t += u[k]
            v[k] = t
        for k in range(i, min(s, K) + 1):
            t += u[k]
            v[k] = t
            t -= u[k - i]
    print(v[K] % (10 ** 9 + 7))

main()

Submission status: Accepted

Solution code:
def candies(n,k,a): 
    mod=10**9+7
    dp=[[ 0 for _ in range(k+1) ] for _ in range(n+1)]
    for i in range(n+1):
        dp[i][0]=1
    for i in range(n):
        for j in range(1,k+1):
            if j-1-a[i]>=0: 
                dp[i+1][j]=( dp[i+1][j-1]+dp[i][j]-dp[i][j-1-a[i]] +mod ) % mod
            else:
                dp[i+1][j]=( dp[i+1][j-1]+dp[i][j] ) %mod
    print(dp[n][k])


def main():
    n,k = map(int,input().split())
    a = list(map(int,input().split()))
    candies(n,k,a)

if __name__ == "__main__":
    main()

Submission status: Accepted

Solution code:
import sys
n, k = [int(i) for i in sys.stdin.readline().split()]
a_ls = [int(i) for i in sys.stdin.readline().split()]
memo_ls = [[0 for i in range(k+1)] for j in range(n)]
memo_ls[0][0] = 1
for i in range(n):
    for j in range(k+1):
        if i == 0:
            if j <= a_ls[i]:
                memo_ls[i][j] = 1
            continue
        if j == 0:
            memo_ls[i][j] = 1
            continue
        if a_ls[i] < j:
            memo_ls[i][j] = (- memo_ls[i-1][max(0,j-a_ls[i]-1)] + memo_ls[i][j-1] + memo_ls[i-1][j]) % (10**9+7) #sum(memo_ls[i-1][max(0,j - a_ls[i]):j+1]) % (10**9 + 7)
        else:
            memo_ls[i][j] = (memo_ls[i][j-1] + memo_ls[i-1][j]) % (10**9+7)
print(memo_ls[-1][-1])

Submission status: Accepted

Solution code:
import sys
read = sys.stdin.buffer.read
readline = sys.stdin.buffer.readline
readlines = sys.stdin.buffer.readlines

import numpy as np

N,K,*A = map(int,read().split())

MOD = 10**9 + 7

"""
・次々と(1+x+...+x^a)をかけていく
・(1-x^{a+1})をかけたあと(1-x)^{-1}をかける
"""

dp = np.zeros(K+10,np.int64)
dp[0] = 1
for a in A:
    dp[(a+1):] -= dp[:-(a+1)].copy()
    dp %= MOD

for _ in range(N):
    np.cumsum(dp,out=dp); dp %= MOD

answer = dp[K]
print(answer)

Submission status: Accepted

Solution code:
import sys
sys.setrecursionlimit(2147483647)
INF=float("inf")
MOD=10**9+7
input=lambda :sys.stdin.readline().rstrip()
def resolve():
    n,k=map(int,input().split())
    A=list(map(int,input().split()))
    dp=[0]*(k+1)
    dp[k]=1
    for i in range(n):
        ndp=[0]*(k+1)
        # dpの累積和を取っておく
        S=[0]*(k+2)
        for j in range(k+1):
            S[j+1]=S[j]+dp[j]
            S[j+1]%=MOD
        # iterete
        for j in range(k+1):
            ndp[j]=S[min(j+A[i]+1,k+1)]-S[j]
            ndp[j]%=MOD
        dp=ndp
    print(dp[0])
resolve()

Submission status: Accepted

Solution code:
n,k = map(int,input().split())
a = list(map(int,input().split()))
mod = 10**9 + 7
dp = [[0 for _ in range(k+1)] for i in range(n)]

for i in range(n):
    dp[i][0] = 1

sum_a = []
now = 0
for i in a:
    now += i
    sum_a.append(now)
#print(sum_a)
for i in range(n):
    for j in range(1,k+1):
        if sum_a[i] < j:
            dp[i][j] = 0
            break
        dp[i][j] = dp[i][j-1]
        if i > 0:
            dp[i][j] += dp[i-1][j]
            if j>a[i]:
                dp[i][j] -= dp[i-1][j-a[i]-1]

        dp[i][j] %= mod
        #print(i+1,j,dp[i][j])


print(dp[n-1][k])

Submission status: Accepted

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

# dp[i][j]:i番の子まででj個の飴を分け合う方法の数。
# iの場合、i-1番目の子までj-k個の飴を分け合い、残りのk個をi+1番目の子に与えると考えると
# dp[i][j] = dp[i-1][j] + dp[i-1][j-1] + dp[i-1][j-2] + ... + dp[i-1][j-ai]
# となる。この式のjをj-1とすると
# dp[i][j-1] = dp[i-1][j-1] + dp[i-1][j-2] + dp[i-1][j-3] + ... + dp[i-1][j-1-ai]
# となり、2つの式を引き算すると
# dp[i][j] = dp[i][j-1] + dp[i-1][j] - dp[i-1][j-1-ai]
# となる。
# j-1-ai < 0 の場合は
# dp[i][j]   = dp[i-1][j] + dp[i-1][j-1] + dp[i-1][j-2] + ... + dp[i-1][0]
# dp[i][j-1] =              dp[i-1][j-1] + dp[i-1][j-2] + ... + dp[i-1][0]
# となるから
# dp[i][j]   = dp[i][j-1] + dp[i-1][j]

dp = [[0 for _ in range(K+1)] for _ in range(N)]

for i in range(K+1):
  if A[0] >= i:
    dp[0][i] = 1
#print(dp)
for i in range(1,N):
  for j in range(K+1):
    if j == 0:
      dp[i][j] = dp[i-1][j]
    elif j-1-A[i] >= 0:
      dp[i][j] = dp[i][j-1] + dp[i-1][j] - dp[i-1][j-1-A[i]]
    else:
      dp[i][j] = dp[i][j-1] + dp[i-1][j]
    dp[i][j] %= MOD
  #print(dp)

print(dp[N-1][K])

Submission status: Accepted

Solution code:
#!/usr/bin/env python3
#EDPC M

import sys
import math
from bisect import bisect_right as br
from bisect import bisect_left as bl
sys.setrecursionlimit(1000000000)
from heapq import heappush, heappop,heappushpop
from collections import defaultdict
from itertools import accumulate
from collections import Counter
from collections import deque
from operator import itemgetter
from itertools import permutations
mod = 10**9 + 7
inf = float('inf')
def I(): return int(sys.stdin.readline())
def LI(): return list(map(int,sys.stdin.readline().split()))

n,k = LI()
a = LI()
dp = [[0]*(k+1) for _ in range(n+1)]
dp[0][0] = 1
for i in range(1,n+1):
    lst = [0] + list(accumulate(dp[i-1]))
    for j in range(k+1):
        if j - a[i-1] < 0:
            dp[i][j] += lst[j+1] - lst[0]
            dp[i][j] %= mod
        else:
            dp[i][j] += lst[j+1] - lst[j-a[i-1]]
            dp[i][j] %= mod
print(dp[-1][-1])

Submission status: Accepted

Solution code:
from itertools import accumulate

def f(k, A):
    n = len(A)

    mod = int(1e9 + 7)
    dp = [0] * (k + 1)

    # init for last column
    for j in range(min(A[n - 1], k) + 1):
        dp[j] = 1

    for i in range(n - 2, -1, -1):
        dp = list(accumulate(dp))
        a = A[i]
        dp[a+1:] = [x - y for x, y in zip(dp[a + 1:], dp)]
        dp = [x % mod for x in dp]

    return dp[-1] % mod

assert f(4, [2, 3]) == 2
assert f(4, [1, 2, 3]) == 5
assert f(10, [9]) == 0
assert f(0, [0, 0]) == 1
assert f(100000, [100000, 100000, 100000, 100000]) == 665683269

n, k = map(int, input().split())
A = list(map(int, input().split()))
ans = f(k, A)
print(ans)

Submission status: Accepted

Solution code:
mod=1000000007
# from https://ikatakos.com/pot/programming_algorithm/contest_history/atcoder/2019/0106_educational_dp
from itertools import accumulate, starmap
from operator import sub

n,k=map(int,input().split())
a=[int(i) for i in input().split()]
dp=[0]*(k+1)
dp[0]=1
for i in a:
    dp=list(accumulate(dp))
    dp[i+1:]=starmap(sub,zip(dp[i+1:],dp[:-i-1]))
    dp=[x%mod for x in dp]
print(dp[k])

Submission status: Accepted

Solution code:
mod = 10 ** 9 + 7

n, k = map(int, input().split())
a = tuple(int(x) for x in input().split())

d = tuple([0] * (k + k + 1) for _ in range(n + 1))
d[0][0] = 1

# もらう
for pind, lim in enumerate(a, 1):
    csum = [0]
    t = 0
    for cnt in range(k + 1):
        t += d[pind - 1][cnt]
        csum.append(t)
    for distributed in range(k + 1):
        d[pind][distributed] = (csum[distributed + 1] - csum[max(0, distributed - lim)]) % mod

print(d[n][k])

Submission status: Accepted

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

N,K = map(int,input().split())
A = list(map(int,input().split()))

dp = [[0]*(K+1) for i in range(N+1)]

dp[0][0] = 1

cum = [0]*(K+2)

for i in range(1,N+1) :
    cum[0] = 1
    for j in range(1,K+2) :
        cum[j] = (cum[j-1] + dp[i-1][j-1]) % (10**9+7)
    for j in range(K+1) :
        dp[i][j] = (cum[j+1] - cum[max(0, j-A[i-1])] + (10**9+7)) % (10**9+7)

ans = dp[N][K]

print(ans)
#print(dp)