91Ó°ÊÓ

Find the exponential generating function for each of the following sequences. a) \(1,-1,1,-1,1,-1, \ldots\) b) \(1,2,2^{2}, 2^{3}, 2^{4}, \ldots\) c) \(1,-a, a^{2},-a^{3}, a^{4}, \ldots, \quad a \in \mathbf{R}\) d) \(1, a^{2}, a^{4}, a^{6}, \ldots, \quad a \in \mathbf{R}\) e) \(a, a^{3}, a^{5}, a^{7}, \ldots, \quad a \in \mathbf{R}\) f) \(0,1,2(2), 3\left(2^{2}\right), 4\left(2^{3}\right), \ldots\)

Determine the sequence generated by each of the following exponential generating functions. a) \(f(x)=3 e^{3 x}\) b) \(f(x)=6 e^{5 x}-3 e^{2 x}\) c) \(f(x)=e^{x}+x^{2}\) d) \(f(x)=e^{2 x}-3 x^{3}+5 x^{2}+7 x\) e) \(f(x)=1 /(1-x)\) f) \(f(x)=3 /(1-2 x)+e^{x}\)

Find the generating function for the number of integer solutions of a) \(2 w+3 x+5 y+7 z=n, \quad 0 \leq w, x, y, z\) b) \(2 w+3 x+5 y+7 z=n, \quad 0 \leq w, \quad 4 \leq x, y, \quad 5 \leq z\)

What is the generating function for the number of partitions of \(n \in \mathbf{N}\) into summands that (a) cannot occur more than five times; and (b) cannot exceed 12 and cannot occur more than five times?

a) Find the generating function for the sequence \(0,1,3,6\), \(10,15, \ldots\) (where \(1,3,6,10,15, \ldots\) are the triangular numbers of Example 4.5). b) For \(n \in \mathbf{Z}^{+}\), determine a formula for the sum of the first \(n\) triangular numbers.

Determine the generating function for the number of partitions of \(n \in \mathbf{N}\) where 1 occurs at most once, 2 occurs at most twice, 3 at most thrice, and, in general, \(k\) occurs at most \(k\) times, for every \(k \in \mathbf{Z}^{+}\).

How many 20 -digit quaternary \((0,1,2,3)\) sequences are there where: (a) There is at least one 2 and an odd number of 0 's? (b) No symbol occurs exactly twice? (c) No symbol occurs exactly three times? (d) There are exactly two 3 's or none at all?

How can Mary split up 12 hamburgers and 16 hot dogs among her sons Richard, Peter, Christopher, and James in such a way that James gets at least one hamburger and three hot dogs, and each of his brothers gets at least two hamburgers but at most five hot dogs?