$$1 + x + x^2 + x^3 + \cdots$$

$$1 + x + 2x^2 + 4x^3 + \cdots$$

$$1 + x + 3x^2 + 9x^3 + \cdots$$

$$1 + x + 4x^2 + 16x^3 + \cdots$$

$$M_n = M_{n-1} + M_{n-2}$$

$$M_n = M_{n-1} + 2M_{n-2}$$

$$M_n = M_{n-1} + 3M_{n-2}$$

$$M_n = M_{n-1} + 4M_{n-2}$$

5

10

15

20

$$M_n \sim \frac{1}{n^{3/2}}\left(\frac{3}{2}\right)^n$$

$$M_n \sim \frac{1}{n^{5/2}}\left(\frac{3}{2}\right)^n$$

$$M_n \sim \frac{1}{n^{7/2}}\left(\frac{3}{2}\right)^n$$

$$M_n \sim \frac{1}{n^{9/2}}\left(\frac{3}{2}\right)^n$$

Counting lattice paths

Counting triangulations

Counting permutations

Counting graphs

$$M_n = 2C_n$$

$$M_n = 3C_n$$

$$M_n = 4C_n$$

$$M_n = 5C_n$$

$$M_n = F_{2n}$$

$$M_n = F_{2n+1}$$

$$M_n = F_{2n+2}$$

$$M_n = F_{2n+3}$$

$$M_n = S(n,n)$$

$$M_n = S(n,n+1)$$

$$M_n = S(n+1,n)$$

$$M_n = S(n+1,n+1)$$

$$M_n = D(n,n)$$

$$M_n = D(n,n+1)$$

$$M_n = D(n+1,n)$$

$$M_n = D(n+1,n+1)$$

$$M_n = S_n$$

$$M_n = S_{n+1}$$

$$M_n = S_{n+2}$$

$$M_n = S_{n+3}$$

$$M_n = N_n$$

$$M_n = N_{n+1}$$

$$M_n = N_{n+2}$$

$$M_n = N_{n+3}$$

$$M_n = CM_n$$

$$M_n = CM_{n+1}$$

$$M_n = CM_{n+2}$$

$$M_n = CM_{n+3}$$

$$M_n = T_n$$

$$M_n = T_{n+1}$$

$$M_n = T_{n+2}$$

$$M_n = T_{n+3}$$

$$M_n = A_n$$

$$M_n = A_{n+1}$$

$$M_n = A_{n+2}$$

$$M_n = A_{n+3}$$

$$M_n = B_n$$

$$M_n = B_{n+1}$$

$$M_n = B_{n+2}$$

$$M_n = B_{n+3}$$