What is the generating function for the sequence 1, 6, 16, 216…?

To find the generating function for the sequence 1, 6, 16, 216..., we can follow these steps:

Step 1: Write down the sequence in a general form. The given sequence can be written as 1, 6, 16, 216, ... = a0, a1, a2, a3, ...

Step 2: Define the generating function. Let's denote the generating function as G(x).

Step 3: Express the generating function in terms of the sequence. The generating function G(x) can be expressed as: G(x) = a0 + a1x + a2x^2 + a3*x^3 + ...

Step 4: Simplify the generating function. Substituting the values of the sequence, we have: G(x) = 1 + 6x + 16x^2 + 216*x^3 + ...

Step 5: Determine the pattern in the sequence. Looking at the sequence, we can observe that each term is a power of x multiplied by a coefficient. The coefficients seem to follow a pattern: 1, 6, 16, 216.

Step 6: Find the general formula for the coefficients. To find the general formula for the coefficients, we can notice that the nth term can be expressed as n^3. Therefore, the general formula for the coefficients is n^3.

Step 7: Rewrite the generating function using the general formula. Using the general formula for the coefficients, we can rewrite the generating function as: G(x) = 1 + 6x + 16x^2 + 216*x^3 + ...

Step 8: Simplify the generating function. Simplifying further, we have: G(x) = 1 + 6x + 16x^2 + 216*x^3 + ...

Therefore, the generating function for the sequence 1, 6, 16, 216... is G(x) = 1 + 6x + 16x^2 + 216*x^3 + ...