7.4: Generating Functions g(x) = h_{0} + h_{1}x + h_{2}x^{2} + .... is the generating function for the sequence h_{0}, h_{1}, h_{2}, ..... Ex: The generating fn for the sequence 2, 3, 4, 0, 0, 0, ... is g(x) = 2 + 3x + 4x^{2} Ex: The generating function for the sequence 1, 1, 1, ... is g(x) = 1 + x + x^{2} + ... = \frac{1}{1−x} Ex: The generating function for the sequence

( \begin{matrix}m \\ 0 \end{matrix} ), ( \begin{matrix}m \\ 1 \end{matrix} ), ( \begin{matrix}m \\ 2 \end{matrix} ), ..., ( \begin{matrix}m \\ m \end{matrix} ) is

g(x) = ( \begin{matrix}m \\ 0 \end{matrix} ) + ( \begin{matrix}m \\ 1 \end{matrix} ) x + ( \begin{matrix}m \\ 2 \end{matrix} ) x^{2} + ... ( \begin{matrix}m \\ m \end{matrix} ) x^{m} = (1 + x)^{m}

Ex: Suppose α ∈ R. The generating function for the sequence

( \begin{matrix}α \\ 0 \end{matrix} ), ( \begin{matrix}α \\ 1 \end{matrix} ), ( \begin{matrix}α \\ 2 \end{matrix} ), ..., is

g(x) = ( \begin{matrix}α \\ 0 \end{matrix} ) + ( \begin{matrix}α \\ 1 \end{matrix} ) x + ( \begin{matrix}α \\ 2 \end{matrix} ) x^{2} + ... = (1 + x)^{α}

Ex: Let h_{n} = number of nonnegative solutions to e_{1} + e_{2} + ... + e_{k} = n Thus h_{n} =

Thus g(x) =

Suppose a multiset consisting of integers between 0 and 5 inclusive of size k must contain the following: even number of 0’s odd number of 1’s three or four 2’s the number of 3’s is a multiple of five between zero to four (inclusive) 4’s zero or one 5 Find the number of multisets of size k. Find the number of multisets of size 100.