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This works! We obtain finally
Crazy Dice 7
a
2
2 2
z zà 1 + z + z )(1 + zð( 1 − z + z )
4
3
6
5
z + z + z + z + z + z 8
and
b 2 2 3 4
z zà 1 + z + z )(1 + zð z + 2z + 2z + z .
Translating back, the crazy dice are 1,3,4,5,6,8 and 1,2,2,3,3,4.

Now we introduce the notion of the representation function. So,
suppose there is a set A of nonnegative integers and that we wish to
express the number of ways in which a given integer n can be written
as the sum of two of them. The trouble is that we must decide on
conventions. Does order count? Can the two summands be equal?
Therefore we introduce three representation functions.

r(n) #{(a, a ) : a, a ∈ A, n a + a };

So here order counts, and they can be equal;

r + (n) #{(a, a ) : a, a ∈ A, a ≤ a ,è a + a },
order doesn’t count, and they can be equal;

r − (n) #{(a, a ) : a, a ∈ A, a < a ,è a + a },

order doesn’t count, and they can’t be equal. In terms of the generat-
a
ing function for the set A, namely, A(z) z , we can express
a∈A
the generating functions of these representation functions.
The simplest is that of r(n), where obviously

2
n
r(n)z A (z). (7)
2
2
To deal with r − (n), we must subtract A(z ) from A (z) to remove

the case of a a and then divide by 2 to remove the order. So here
1
n 2 2
r − (n)z [A (z) − A(z )]. à 8)
2