Tushar joshi perfect string

[tusharjoshi4531]

My solution to Perfect string.

f(i) = number of perfect string of length 2 * i
g(i) = number of perfect strings of length 2 * i which cannot be partitioned into contiguous perfect strings.
c = number pf charecters

F(x) = sum(f(i), x^i) for i = 0 - inf
G(x) = sum(g(i), x^i) for i = 0 - inf

derived equations -
F(x) = F(x)G(x) + 1;
G(x) = cxF(x) - G(x)F(x) / c

after solving
F(x) = [1 / (1 - xc^2) ]- c * [(1 - sqrt(1 - 4(c - 1)x) / 2 (c - 1) (x)] * [(c - 1)x / (1 - xc^2)]

after maclaurin expansion
f(n) = c ^ 2n - c*(c - 1) sum(Catalan(i) * (c - 1) ^ i * (c) ^ (2 * (n - 1- i)), i = 0 - n - 1)

solution implemented in O(n) time and O(n) space

Time of execution = 2.9 s.