Количество разбиений строки на группы различной длины

from  math import ceil, factorial
 
 
n = 70
m = 2
 
r = 0
for k in range(ceil(n / m), n + 1):
    for j in range(min(k, ceil((n - k - k) / m)) + 1):
        a = factorial(k) / factorial(k - j) / factorial(j)
 
        i = k - 1
        z = n - k - m*j - 1
        b = factorial(z) / factorial(z - i) / factorial(i)
 
        r += -1**j * a * b
        
 
print(r)

from  math import ceil, factorial
 
 
n = 10
m = 2
 
r = 0
for k in range(ceil(n / m), n + 1):
    for i in range(int((n - k) / m) + 1):
 
        a = -1**i
 
        b = factorial(k) / factorial(k - i) / factorial(i)
 
        x = k - 1
        y = n - m*i - 1
        c = factorial(y) / factorial(y - x) / factorial(x)
 
        r += a * b * c
        
print(r)

from  math import ceil, factorial
 
 
def BinomKoef(n, k):
    result = 1
    for i in range(1, k + 1):
        result *= (n - i + 1) / i
    return(result)
 
def BinomKoef1(n, k):
    return factorial(n) // factorial(n - k) // factorial(k)
 
def DoubleSumma(n, m):
    result = 0
    for k in range(ceil(n / m), n + 1):
        for i in range(int((n - k) / m) + 1):
            result += (-1)**i * BinomKoef(k, i) * BinomKoef(n - m * i - 1, k - 1)
    return(result)
        
print('n , m: result')
print(' 1, 2:', DoubleSumma(1, 2))
print(' 2, 2:', DoubleSumma(2, 2))
print(' 3, 2:', DoubleSumma(3, 2))
print(' 4, 2:', DoubleSumma(4, 2))
print(' 5, 2:', DoubleSumma(5, 2))
print('10, 2:', DoubleSumma(10, 2))
 
print('20, 2:', DoubleSumma(20, 2), 'Ошибка:', 10946 - DoubleSumma(20, 2))
print('70, 2:', DoubleSumma(70, 2), 'Ошибка:', 308061521170129 - DoubleSumma(70, 2))