Buggy python submission (verdict: Time Limit Exceeded):

import numpy as np
import copy
# N,K = map(int,input().split())
# V = list(map(int,input().split()))
N = 17
K = 100
V = [i for i in range(N)]

V = [0 for i in range(N)] + V + [0 for i in range(N)] 

dp = {}

# discardフラグを立てて，一度捨てると取れないように．
def search(start, end, k, own, discard=False):
    if k==K:
        return sum(own)
    key = tuple(own)
    if key in dp:
        return dp[key]
    val_Default = sum(own)
    val_A,val_B = 0,0
    if end-start >= 0 and not discard:
        val_A = search(start+1, end, k+1, own + [V[start]])
        val_B = search(start, end-1, k+1, own + [V[end]])
    if len(own) == 0 and discard:
        return val_Default
    val_Cs = []
    val_Ds = []
    for i,o in enumerate(own):
        own_new = copy.copy(own)
        own_new.pop(i)
        val_Cs.append(search(-1, -1, k+1, own_new, discard=True) )
        # val_Cs.append(search(start-1, end, k+1, own_new, discard=True) )
        # val_Ds.append(search(start, end+1, k+1, own_new, discard=True) )
    # print([val_Default, val_A, val_B], val_Cs , val_Ds)
    # dp[key] = max([val_Default, val_A, val_B] + val_Cs + val_Ds)
    dp[key] = max([val_Default, val_A, val_B] + val_Cs)
    # print(own, [val_Default, val_A, val_B], val_Cs, val_Ds)
    return dp[key]

r = search(N, 2*N-1, 0, [])
print(r)



Fixed submission:
N,K = map(int,input().split())
V = list(map(int,input().split()))

sumval = 0
for l in range(min(K+1,N+1)):
    for r in range(min(K+1-l,N-l+1)):
        sV = sorted(V[:l]+ V[N-r:])
        # i_div = 0
        # for i_div in range(len(sV)):
        #     if sV[i_div] > 0:
        #         break
        # for sute in range(min(K-r-l+1,len(sV),i_div+1)):
        for sute in range(min(K-r-l+1,len(sV))):
            c = sum(sV[sute:])
            sumval = max(c,sumval)

print(sumval)

Buggy python submission (verdict: Runtime Error):
import numpy as np
import copy
N,K = map(int,input().split())
V = list(map(int,input().split()))
# N = 17
# K = 100
# V = [i for i in range(N)]


V = [0 for i in range(N)] + V + [0 for i in range(N)] 


dp = {}

# discardフラグを立てて，一度捨てると取れないように．
def search(start, end, k, own, discard=False):
    if k==K:
        return sum(own)
    key = tuple(own)
    if key in dp:
        return dp[key]
    val_Default = sum(own)
    val_A,val_B = 0,0
    if end-start >= 0 and not discard:
        val_A = search(start+1, end, k+1, own + [V[start]])
        val_B = search(start, end-1, k+1, own + [V[end]])
    if len(own) == 0 and discard:
        return val_Default
    val_Cs = []
    val_Ds = []
    for i,o in enumerate(own):
        own_new = copy.copy(own)
        own_new.pop(i)
        val_Cs.append(search(-1, -1, k+1, own_new, discard=True) )
        # val_Cs.append(search(start-1, end, k+1, own_new, discard=True) )
        # val_Ds.append(search(start, end+1, k+1, own_new, discard=True) )
    # print([val_Default, val_A, val_B], val_Cs , val_Ds)
    # dp[key] = max([val_Default, val_A, val_B] + val_Cs + val_Ds)
    dp[key] = max([val_Default, val_A, val_B] + val_Cs)
    # print(own, [val_Default, val_A, val_B], val_Cs, val_Ds)
    return dp[key]

r = search(N, 2*N-1, 0, [])
print(r)



Fixed submission:
N,K = map(int,input().split())
V = list(map(int,input().split()))

sumval = 0
for l in range(min(K+1,N+1)):
    for r in range(min(K+1-l,N-l+1)):
        sV = sorted(V[:l]+ V[N-r:])
        # i_div = 0
        # for i_div in range(len(sV)):
        #     if sV[i_div] > 0:
        #         break
        # for sute in range(min(K-r-l+1,len(sV),i_div+1)):
        for sute in range(min(K-r-l+1,len(sV))):
            c = sum(sV[sute:])
            sumval = max(c,sumval)

print(sumval)