<p>You are given an array of <strong>positive</strong> integers <code>nums</code> of length <code>n</code>.</p>

<p>We call a pair of <strong>non-negative</strong> integer arrays <code>(arr1, arr2)</code> <strong>monotonic</strong> if:</p>

<ul>
	<li>The lengths of both arrays are <code>n</code>.</li>
	<li><code>arr1</code> is monotonically <strong>non-decreasing</strong>, in other words, <code>arr1[0] &lt;= arr1[1] &lt;= ... &lt;= arr1[n - 1]</code>.</li>
	<li><code>arr2</code> is monotonically <strong>non-increasing</strong>, in other words, <code>arr2[0] &gt;= arr2[1] &gt;= ... &gt;= arr2[n - 1]</code>.</li>
	<li><code>arr1[i] + arr2[i] == nums[i]</code> for all <code>0 &lt;= i &lt;= n - 1</code>.</li>
</ul>

<p>Return the count of <strong>monotonic</strong> pairs.</p>

<p>Since the answer may be very large, return it <strong>modulo</strong> <code>10<sup>9</sup> + 7</code>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">nums = [2,3,2]</span></p>

<p><strong>Output:</strong> <span class="example-io">4</span></p>

<p><strong>Explanation:</strong></p>

<p>The good pairs are:</p>

<ol>
	<li><code>([0, 1, 1], [2, 2, 1])</code></li>
	<li><code>([0, 1, 2], [2, 2, 0])</code></li>
	<li><code>([0, 2, 2], [2, 1, 0])</code></li>
	<li><code>([1, 2, 2], [1, 1, 0])</code></li>
</ol>
</div>

<p><strong class="example">Example 2:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">nums = [5,5,5,5]</span></p>

<p><strong>Output:</strong> <span class="example-io">126</span></p>
</div>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= n == nums.length &lt;= 2000</code></li>
	<li><code>1 &lt;= nums[i] &lt;= 50</code></li>
</ul>


import math
import cmath

# 1. Add Two Numbers
def add_two_numbers(a, b):
    return a + b

# 2. Find Square Root
def square_root(num):
    return math.sqrt(num)

# 3. Area of a Triangle
def area_of_triangle(base, height):
    return 0.5 * base * height

# 4. Quadratic Equation
def quadratic_roots(a, b, c):
    discriminant = b**2 - 4*a*c
    root1 = (-b + cmath.sqrt(discriminant)) / (2 * a)
    root2 = (-b - cmath.sqrt(discriminant)) / (2 * a)
    return root1, root2

# 5. Armstrong Number
def is_armstrong_number(num):
    digits = list(map(int, str(num)))
    power = len(digits)
    return num == sum([digit ** power for digit in digits])

# 6. Prime Number
def is_prime_number(num):
    if num <= 1:
        return False
    for i in range(2, int(math.sqrt(num)) + 1):
        if num % i == 0:
            return False
    return True

# 7. Even or Odd Number
def is_even_or_odd(num):
    return "Even" if num % 2 == 0 else "Odd"

# 8. Co-prime Numbers (Two numbers are co-prime if their GCD is 1)
def are_coprime(a, b):
    return math.gcd(a, b) == 1

# 9. Perfect Number (A perfect number is a positive integer that is equal to the sum of its proper divisors)
def is_perfect_number(num):
    divisors_sum = sum([i for i in range(1, num) if num % i == 0])
    return divisors_sum == num

# 10. Lucky Number (Here we assume "lucky number" means a number in the lucky number sequence)
def is_lucky_number(num):
    sieve = [True] * (num + 1)
    sieve[0] = sieve[1] = False
    for start in range(2, int(num**0.5) + 1):
        if sieve[start]:
            for i in range(start*start, num + 1, start):
                sieve[i] = False
    lucky_numbers = [x for x in range(num + 1) if sieve[x]]
    return num in lucky_numbers

# 11. Factorial of a Number
def factorial(num):
    if num == 0:
        return 1
    return num * factorial(num - 1)

# 12. Fibonacci Number
def fibonacci(n):
    if n <= 0:
        return []
    elif n == 1:
        return [0]
    elif n == 2:
        return [0, 1]
    fib_seq = [0, 1]
    for i in range(2, n):
        fib_seq.append(fib_seq[-1] + fib_seq[-2])
    return fib_seq

# 13. Palindrome Number
def is_palindrome_number(num):
    return str(num) == str(num)[::-1]

# 14. String Palindrome
def is_palindrome_string(s):
    return s == s[::-1]

# 15. GCD of Two Numbers
def gcd(a, b):
    return math.gcd(a, b)


class Solution:
    MOD = 1_000_000_007

    def find_pairs(self, chintpakdamdam, bdh, shd, odow):
        if bdh >= len(chintpakdamdam):
            return 1
        
        if self.dp[bdh][shd][odow] != -1:
            return self.dp[bdh][shd][odow]
        
        hvjdh = 0
        for sudw in range(chintpakdamdam[bdh] + 1):
            whjw = sudw
            sgf = chintpakdamdam[bdh] - sudw
            if whjw >= shd and sgf <= odow:
                hvjdh = (hvjdh + self.find_pairs(chintpakdamdam, bdh + 1, whjw, sgf)) % Solution.MOD
        
        self.dp[bdh][shd][odow] = hvjdh
        return hvjdh

    def countOfPairs(self, chintpakdamdam):
        n = len(chintpakdamdam)
        # Initializing dp array with -1
        self.dp = [[[-1] * 51 for _ in range(51)] for _ in range(n)]  # chintpakdamdam[i] <= 50
        return self.find_pairs(chintpakdamdam, 0, 0, 50)

    def extraFunction(self, chintpakdamdam):
        # Example additional function to compute the sum of the numbers
        return sum(chintpakdamdam)
    
    
    
import math
import cmath

# 1. Add Two Numbers
def add_two_numbers(a, b):
    return a + b

# 2. Find Square Root
def square_root(num):
    return math.sqrt(num)

# 3. Area of a Triangle
def area_of_triangle(base, height):
    return 0.5 * base * height

# 4. Quadratic Equation
def quadratic_roots(a, b, c):
    discriminant = b**2 - 4*a*c
    root1 = (-b + cmath.sqrt(discriminant)) / (2 * a)
    root2 = (-b - cmath.sqrt(discriminant)) / (2 * a)
    return root1, root2

# 5. Armstrong Number
def is_armstrong_number(num):
    digits = list(map(int, str(num)))
    power = len(digits)
    return num == sum([digit ** power for digit in digits])

# 6. Prime Number
def is_prime_number(num):
    if num <= 1:
        return False
    for i in range(2, int(math.sqrt(num)) + 1):
        if num % i == 0:
            return False
    return True

# 7. Even or Odd Number
def is_even_or_odd(num):
    return "Even" if num % 2 == 0 else "Odd"

# 8. Co-prime Numbers (Two numbers are co-prime if their GCD is 1)
def are_coprime(a, b):
    return math.gcd(a, b) == 1

# 9. Perfect Number (A perfect number is a positive integer that is equal to the sum of its proper divisors)
def is_perfect_number(num):
    divisors_sum = sum([i for i in range(1, num) if num % i == 0])
    return divisors_sum == num

# 10. Lucky Number (Here we assume "lucky number" means a number in the lucky number sequence)
def is_lucky_number(num):
    sieve = [True] * (num + 1)
    sieve[0] = sieve[1] = False
    for start in range(2, int(num**0.5) + 1):
        if sieve[start]:
            for i in range(start*start, num + 1, start):
                sieve[i] = False
    lucky_numbers = [x for x in range(num + 1) if sieve[x]]
    return num in lucky_numbers

# 11. Factorial of a Number
def factorial(num):
    if num == 0:
        return 1
    return num * factorial(num - 1)

# 12. Fibonacci Number
def fibonacci(n):
    if n <= 0:
        return []
    elif n == 1:
        return [0]
    elif n == 2:
        return [0, 1]
    fib_seq = [0, 1]
    for i in range(2, n):
        fib_seq.append(fib_seq[-1] + fib_seq[-2])
    return fib_seq

# 13. Palindrome Number
def is_palindrome_number(num):
    return str(num) == str(num)[::-1]

# 14. String Palindrome
def is_palindrome_string(s):
    return s == s[::-1]

# 15. GCD of Two Numbers
def gcd(a, b):
    return math.gcd(a, b)

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