Problem:
 Score : 1200 points    Problem Statement Snuke has decided to play with N cards and a deque (that is, a double-ended queue).
Each card shows an integer from 1 through N , and the deque is initially empty.  Snuke will insert the cards at the beginning or the end of the deque one at a time, in order from 1 to N .
Then, he will perform the following action N times: take out the card from the beginning or the end of the deque and eat it.  Afterwards, we will construct an integer sequence by arranging the integers written on the eaten cards, in the order they are eaten. Among the sequences that can be obtained in this way, find the number of the sequences such that the K -th element is 1 . Print the answer modulo 10^{9} + 7 .      Constraints  1 ≦ K ≦ N ≦ 2{,}000         Input The input is given from Standard Input in the following format:  N  K       Output Print the answer modulo 10^{9} + 7 .        Sample Input 1 2 1      Sample Output 1 1  There is one sequence satisfying the condition: 1,2 . One possible way to obtain this sequence is the following:   Insert both cards, 1 and 2 , at the end of the deque.  Eat the card at the beginning of the deque twice.        Sample Input 2 17 2      Sample Output 2 262144       Sample Input 3 2000 1000      Sample Output 3 674286644

Solution (Python):
import sys
import numpy as np
import numba
from numba import njit, b1, i4, i8
from numba.types import Omitted

read = sys.stdin.buffer.read
readline = sys.stdin.buffer.readline
readlines = sys.stdin.buffer.readlines

MOD = 1_000_000_007

@njit((i8, i8), cache=True)
def main(N, K):
    K -= 1
    U = 2010
    C = np.zeros((U, U), np.int64)
    C[0, 0] = 1
    for n in range(1, U):
        C[n] += C[n - 1]
        C[n, 1:] += C[n - 1, :-1]
        C[n] %= MOD

    dp = np.zeros((N + 10, N + 10), np.int64)
    dp[0, 0] = 1
    for n in range(1, N + 10):
        dp[n, n] = (dp[n - 1, n - 1] + dp[n - 1, n]) % MOD
        for m in range(n + 1, N + 10):
            dp[n, m] = (dp[n - 1, m] + dp[n, m - 1]) % MOD
    for n in range(N + 9, 0, -1):
        dp[n] -= dp[n - 1]
        dp[n] %= MOD
    ans = 0
    if K == N - 1:
        return dp[:N, N - 1].sum() % MOD
    for r in range(N - K, N + 1):
        x = C[r - 2, N - K - 2]
        a = N - r
        b = K - a
        if b == 0:
            x = x * (dp[:a + 1, a].sum() % MOD) % MOD
        else:
            x *= (dp[1:a + 2, a + 1] * C[b - 1:a + b, b - 1][::-1] %
                  MOD).sum() % MOD
            x %= MOD
        ans += x
    ans %= MOD
    for i in range(N - K - 2):
        ans = ans * 2 % MOD
    return ans % MOD

N, K = map(int, read().split())
print(main(N, K))

Solution (Python):
import sys
import numpy as np

read = sys.stdin.buffer.read
readline = sys.stdin.buffer.readline
readlines = sys.stdin.buffer.readlines

MOD = 1_000_000_007

def main(N, K):
    K -= 1
    U = 5000
    C = np.zeros((U, U), np.int64)
    C[0, 0] = 1
    for n in range(1, U):
        C[n] += C[n - 1]
        C[n, 1:] += C[n - 1, :-1]
        C[n] %= MOD

    dp = np.zeros((N + 10, N + 10), np.int64)
    dp[0, 0] = 1
    for n in range(1, N + 10):
        dp[n, n] = (dp[n - 1, n - 1] + dp[n - 1, n]) % MOD
        for m in range(n + 1, N + 10):
            dp[n, m] = (dp[n - 1, m] + dp[n, m - 1]) % MOD
    for n in range(N + 9, 0, -1):
        dp[n] -= dp[n - 1]
        dp[n] %= MOD
    ans = 0
    if K == N - 1:
        return dp[:N, N - 1].sum() % MOD
    for r in range(N - K, N + 1):
        x = C[r - 2, N - K - 2]
        a = N - r
        b = K - a
        if b == 0:
            x = x * (dp[:a + 1, a].sum() % MOD) % MOD
        else:
            x *= (dp[1:a + 2, a + 1] * C[b - 1:a + b, b - 1][::-1] %
                  MOD).sum() % MOD
            x %= MOD
        ans += x
    ans %= MOD
    for i in range(N - K - 2):
        ans = ans * 2 % MOD
    return ans % MOD

if sys.argv[-1] == 'ONLINE_JUDGE':
    import numba
    from numba.pycc import CC
    i4 = numba.int32
    i8 = numba.int64
    cc = CC('my_module')

    def cc_export(f, signature):
        cc.export(f.__name__, signature)(f)
        return numba.njit(f)

    main = cc_export(main, (i8, i8))
    cc.compile()

from my_module import main

N, K = map(int, read().split())
print(main(N, K))

Solution (Python):
def cmb(n, r, mod):
    if ( r<0 or r>n ):
        return 0
    r = min(r, n-r)
    return g1[n] * g2[r] * g2[n-r] % mod

mod = 10**9+7 
N = 5000
g1 = [1, 1] 
g2 = [1, 1] 
inverse = [0, 1] 

for i in range( 2, N + 1 ):
    g1.append( ( g1[-1] * i ) % mod )
    inverse.append( ( -inverse[mod % i] * (mod//i) ) % mod )
    g2.append( (g2[-1] * inverse[-1]) % mod )

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

if N==1:
    print(1)
elif K==1:
    print(pow(2,N-2,mod))
elif K==N:
    dp=[[0 for i in range(K+1)] for j in range(K)]
    imos=[0]*(K+1)

    dp[0][K]=1
    imos[K]=1

    for i in range(1,K):
        for j in range(K-i,K+1):
            if j==K-i:
                dp[i][j]=(imos[K]-imos[j])%mod
            else:
                dp[i][j]=(dp[i-1][j]+imos[K]-imos[j])%mod
        imos=[dp[i][j] for j in range(K+1)]
        for j in range(1,K+1):
            imos[j]+=imos[j-1]
            imos[j]%=mod
    print(dp[N-1][1])
else:
    dp=[[0 for i in range(K+1)] for j in range(K)]
    imos=[0]*(K+1)

    dp[0][K]=1
    imos[K]=1

    for i in range(1,K):
        for j in range(K-i,K+1):
            if j==K-i:
                dp[i][j]=(imos[K]-imos[j])%mod
            else:
                dp[i][j]=(dp[i-1][j]+imos[K]-imos[j])%mod
        imos=[dp[i][j] for j in range(K+1)]
        for j in range(1,K+1):
            imos[j]+=imos[j-1]
            imos[j]%=mod

    ans=0
    for M in range(N-K+1,N+1):
        id=M-N+K
        ans+=dp[K-1][id]*cmb(M-2,N-K-1,mod)

    ans*=pow(2,N-K-1,mod)
    print(ans%mod)