Buggy python submission (verdict: Time Limit Exceeded):
mod = 998244353
import sys
sys.setrecursionlimit(10**9)
S = input()
N = len(S)
Sum = [0]*(N+1)
for i in range(1, N+1):
    Sum[i] = Sum[i-1]+int(S[i-1])
memo = [[-1]*(N*2+1) for _ in range(N*2+1)]
def dp(i, j):
    if memo[i][j]>=0:
        return memo[i][j]
    n = min(i+j+1, N)
    if i+j == N*2:
        return 1
    res = 0
    s = Sum[n]
    if i<s:
        res+=dp(i+1, j)
    if j<2*n-s:
        res+=dp(i, j+1)
    memo[i][j] = res
    return res%mod
print(dp(0, 0))

Fixed submission:
mod = 998244353
import sys
sys.setrecursionlimit(10**9)
S = input()
N = len(S)
Sum = [0]*(N+1)
for i in range(1, N+1):
    Sum[i] = Sum[i-1]+int(S[i-1])
dp = [[0]*(N*2+1) for _ in range(N*2+1)]
for i in range(N*2+1)[::-1]:
    for j in range(N*2+1)[::-1]:
        if i+j == N*2:
            dp[i][j] = 1
            continue
        n = min(i+j+1, N)
        s = Sum[n]
        if i<s:
            dp[i][j]+=dp[i+1][j]
        if 2*n-s>j:
            dp[i][j]+=dp[i][j+1]
        dp[i][j]%=mod
print(dp[0][0])

Buggy python submission (verdict: Time Limit Exceeded):
mod = 998244353
import sys
sys.setrecursionlimit(10**9)
S = input()
N = len(S)
Sum = [0]*(N+1)
for i in range(1, N+1):
    Sum[i] = Sum[i-1]+int(S[i-1])
from functools import lru_cache
@lru_cache(maxsize=10**8)
def dp(i, j):
    n = min(i+j+1, N)
    if i+j == N*2:
        return 1
    res = 0
    s = Sum[n]
    if i<s:
        res+=dp(i+1, j)
    if j<2*n-s:
        res+=dp(i, j+1)
    return res%mod
print(dp(0, 0))

Fixed submission:
mod = 998244353
import sys
sys.setrecursionlimit(10**9)
S = input()
N = len(S)
Sum = [0]*(N+1)
for i in range(1, N+1):
    Sum[i] = Sum[i-1]+int(S[i-1])
dp = [[0]*(N*2+1) for _ in range(N*2+1)]
for i in range(N*2+1)[::-1]:
    for j in range(N*2+1)[::-1]:
        if i+j == N*2:
            dp[i][j] = 1
            continue
        n = min(i+j+1, N)
        s = Sum[n]
        if i<s:
            dp[i][j]+=dp[i+1][j]
        if 2*n-s>j:
            dp[i][j]+=dp[i][j+1]
        dp[i][j]%=mod
print(dp[0][0])

Buggy python submission (verdict: Runtime Error):
mod = 998244353
S = input()
N = len(S)
Sum = [0]*(N+1)
for i in range(1, N+1):
    Sum[i] = Sum[i-1]+int(S[i-1])
from functools import lru_cache
@lru_cache(maxsize=10**8)
def dp(i, j):
    n = min(i+j+1, N)
    if i+j == N*2:
        return 1
    res = 0
    s = Sum[n]
    if i<s:
        res+=dp(i+1, j)
    if j<2*n-s:
        res+=dp(i, j+1)
    return res%mod
print(dp(0, 0))

Fixed submission:
mod = 998244353
import sys
sys.setrecursionlimit(10**9)
S = input()
N = len(S)
Sum = [0]*(N+1)
for i in range(1, N+1):
    Sum[i] = Sum[i-1]+int(S[i-1])
dp = [[0]*(N*2+1) for _ in range(N*2+1)]
for i in range(N*2+1)[::-1]:
    for j in range(N*2+1)[::-1]:
        if i+j == N*2:
            dp[i][j] = 1
            continue
        n = min(i+j+1, N)
        s = Sum[n]
        if i<s:
            dp[i][j]+=dp[i+1][j]
        if 2*n-s>j:
            dp[i][j]+=dp[i][j+1]
        dp[i][j]%=mod
print(dp[0][0])

Buggy python submission (verdict: Runtime Error):
mod = 998244353
S = input()
N = len(S)
Sum = [0]*(N+1)
for i in range(1, N+1):
    Sum[i] = Sum[i-1]+int(S[i-1])
from functools import lru_cache

@lru_cache(maxsize=1000)
def dp(i, j):
    n = min(i+j+1, N)
    if i+j == N*2:
        return 1
    res = 0
    s = Sum[n]
    if i<s:
        res+=dp(i+1, j)
    if j<2*n-s:
        res+=dp(i, j+1)
    return res%mod
print(dp(0, 0))

Fixed submission:
mod = 998244353
import sys
sys.setrecursionlimit(10**9)
S = input()
N = len(S)
Sum = [0]*(N+1)
for i in range(1, N+1):
    Sum[i] = Sum[i-1]+int(S[i-1])
dp = [[0]*(N*2+1) for _ in range(N*2+1)]
for i in range(N*2+1)[::-1]:
    for j in range(N*2+1)[::-1]:
        if i+j == N*2:
            dp[i][j] = 1
            continue
        n = min(i+j+1, N)
        s = Sum[n]
        if i<s:
            dp[i][j]+=dp[i+1][j]
        if 2*n-s>j:
            dp[i][j]+=dp[i][j+1]
        dp[i][j]%=mod
print(dp[0][0])