The generating function for the sequence (1, 2, 3, 4, \dots) is (\frac{x}{(1-x)^2}) because the coefficient of (x^n) in this generating function is (n), which is the (n)-th term of the sequence.

The generating function for the sequence (1, 1, 2, 3, 5, 8, \dots) is (\frac{x}{1-x-x^2}) because the characteristic equation of the recurrence relation (a_n = a_{n-1} + a_{n-2}) is (x^2 + x + 1 = 0), and the roots of this equation are (\frac{-1 \pm \sqrt{-3}}{2}).

The generating function for the sequence (1, 2, 4, 8, 16, \dots) is (\frac{x}{1-2x}) because the coefficient of (x^n) in this generating function is (2^{n-1}), which is the (n)-th term of the sequence.

The generating function for the sequence (1, 3, 6, 10, 15, \dots) is (\frac{x}{(1-x)^3}) because the coefficient of (x^n) in this generating function is (\frac{n(n+1)}{2}), which is the sum of the first (n) positive integers.

The generating function for the sequence (1, 4, 9, 16, 25, \dots) is (\frac{x}{1-x^2}) because the coefficient of (x^n) in this generating function is (n^2), which is the (n)-th term of the sequence.

The generating function for the sequence (1, 2, 4, 7, 11, \dots) is (\frac{x}{(1-x)^4}) because the coefficient of (x^n) in this generating function is (\frac{n(n+1)(2n+1)}{6}), which is the sum of the first (n) odd positive integers.

The generating function for the sequence (1, 3, 5, 7, 9, \dots) is (\frac{x}{1-x^3}) because the coefficient of (x^n) in this generating function is (2n-1), which is the (n)-th term of the sequence.

The generating function for the sequence (1, 4, 9, 16, 25, \dots) is (\frac{x}{(1-x)^5}) because the coefficient of (x^n) in this generating function is (\frac{n(n+1)(2n+1)(3n^2+3n-1)}{30}), which is the square of the first (n) positive integers.

The generating function for the sequence (1, 2, 6, 24, 120, \dots) is (\frac{x}{1-x-x^2-x^3}) because the characteristic equation of the recurrence relation (a_n = a_{n-1} + a_{n-2} + a_{n-3} + a_{n-4}) is (x^4 + x^3 + x^2 + x + 1 = 0), and the roots of this equation are (\frac{-1 \pm \sqrt{-3}}{2}, \frac{-1 \pm i\sqrt{3}}{2}).

The generating function for the sequence (1, 3, 6, 10, 15, \dots) is (\frac{x}{(1-x)^4}) because the coefficient of (x^n) in this generating function is (\frac{n(n+1)(n+2)}{6}), which is the sum of the first (n) triangular numbers.

The generating function for the sequence (1, 5, 14, 30, 55, \dots) is (\frac{x}{1-x-x^2}) because the characteristic equation of the recurrence relation (a_n = a_{n-1} + a_{n-2} + a_{n-3}) is (x^3 + x^2 + x + 1 = 0), and the roots of this equation are (\frac{-1 \pm \sqrt{-3}}{2}, \frac{-1}{2}).

The generating function for the sequence (1, 4, 10, 20, 35, \dots) is (\frac{x}{(1-x)^5}) because the coefficient of (x^n) in this generating function is (\frac{n(n+1)(2n+1)(3n^2+3n-1)}{30}), which is the sum of the first (n) square numbers.

The generating function for the sequence (1, 2, 5, 12, 22, \dots) is (\frac{x}{1-x-x^2-x^3}) because the characteristic equation of the recurrence relation (a_n = a_{n-1} + a_{n-2} + a_{n-3} + a_{n-4}) is (x^4 + x^3 + x^2 + x + 1 = 0), and the roots of this equation are (\frac{-1 \pm \sqrt{-3}}{2}, \frac{-1 \pm i\sqrt{3}}{2}).

The generating function for the sequence (1, 5, 15, 35, 70, \dots) is (\frac{x}{(1-x)^6}) because the coefficient of (x^n) in this generating function is (\frac{n(n+1)(3n-1)}{6}), which is the sum of the first (n) pentagonal numbers.