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b
0
(t) = (1 − t)
4
b
0
+ 4t(1 − t)
3
b
1
+ 6t
2
(1 − t)
2
b
2
+ 4t
3
(t − 1)
b
3
+ t
4
b
4
,
(13.27)
b
0
(t) = (1 − t)
5
b
0
+ 5t(1 − t)
4
b
1
+ 10t
2
(1 − t)
3
b
2
+ 10t
3
(1 − t)
2
b
3
+ 5t
4
(1 − t)
b
4
+ t
5
b
5
.
(13.28)
Now the pattern is more clear. Each term has a constant coe
cient, a power
of (1 − t), and a power of t. The powers of t are numbered in increasing
order, so
b
i
has a coe
cient t
i
. The powers of (1 − t) follow the opposite
pattern and are numbered in decreasing order.
The pattern for the constant coe
cients is a bit more complicated.
Please permit a brief, but hopefully interesting, detour into combinatorics.
Let's write out the first eight levels in a triangular form to make the pattern
a bit easier to see:
0
1
Pascal's triangle
1
1
1
2
1
2
1
3
1
3
3
1
4
1
4
6
4
1
5
1
5
10
10
5
1
6
1
6
15
20
15
6
1
7
1
7
21
35
35
21
7
1
With the exception of the 1s on the outer edge of the triangle, all other
numbers are the sum of the two numbers above it. You are looking at a
very famous number pattern that has been studied for centuries, known as
the binomial coe
cients because the nth row gives the coe
cients when
expanding the binomial (a+b)
n
. The compulsion to organize these numbers
in a triangular manner like this has struck many people, including the
mathematician and physicist Blaise Pascal (1623-1662).
14
This triangular
arrangement of the binomial coe
cients is known as Pascal's triangle.
15
Binomial coe
cients have a special notation. We can refer to the kth
number on row n in Pascal's triangle (where the indexing starts at 0 for
both n and k) using binomial coe
cient notation as
n
k
Binomial coe
cient
notation
.
14
Yes, he was French, too. He appears in Ian Parberry's PhD adviser tree somewhat
off to the left back 16 generations.
15
In addition to his triangle, Pascal has an SI unit of pressure, a law, a programming
language, and a wager named after him, although the latter two are no longer in serious
use.
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