MATH 681
Problem Set #4
This problem set is due at the beginning of class on
October 22
.
1.
(10 points)
For this problem, it will be helpful to note the following two power series
expansions:
e
x
+
e

x
2
= 1 +
x
2
2!
+
x
4
4!
+
x
6
6!
+
· · ·
e
x

e

x
2
=
x
+
x
3
3!
+
x
5
5!
+
x
7
7!
+
· · ·
(a)
(5 points)
Find an exponential generating function for the sequence
{
a
n
}
of the
number of ways to build an
n
letter string consisting of As, Bs, Cs, and Ds such
that there is at least one A, an even number of Bs, an odd number of Cs, and any
number of Ds.
(b)
(5 points)
Using your exponential generating function, find a formula for
a
n
.
2.
(20 points)
A string is called “excellent” if it consists of any number of As, any
number of Bs, and exactly one C. Let
a
n
represent the number of excellent strings of
length
n
.
(a)
(5 points)
Construct an exponential generating function
g
(
x
) =
∑
∞
n
=0
a
n
x
n
n
!
.
(b)
(5 points)
Using casewise analysis on the first term of an excellent string, find a
recurrence relation, with initial conditions, for
a
n
.
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 Fall '09
 WILDSTROM
 Power Series, Recurrence relation, Generating function, exponential generating function

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