【题解】1st ucup Stage 20: India G - Perfect Strings

考虑卡特兰数 $C_n = \sum_{i=0}^{n-1}C_iC_{n-1-i}$，故有递推式

$$C = xC^2 +1$$

解出卡特兰数递推式：

$$C = \frac{1 - \sqrt{1 - 4x}}{2x}$$

考虑本题的递推式：

$$F_n = \sum_{i=0}^{n-1}\frac{c}{c-1}C_iF_{n-1-i}$$

令 $B = c /(c - 1)$故有

$$F = xBCF + 1 \\ F = \frac{1}{1-xBC}\\ F = \frac{1-\frac{B}{2}-\frac{B}{2}\sqrt{1-4x}}{(-B+1)+B^2x}$$

$\sqrt{1-4x} = 1-2xC$ 令

$$F = \frac{1}{1-B} \cdot \frac{1 - B + xBC}{1-\frac{B^2}{B-1}x}$$