-------------------
1 - x^a. 1 - x^b. 1 - x^c. ...

and the problems of finding the partitions of a number n, and of
determining their number, are the same as those of solving and
enumerating the solutions of the indeterminate equation in positive
integers

ax + by + cz + ... = n.

Euler considered also the question of enumerating the solutions of the
indeterminate simultaneous equation in positive integers

ax + by + cz + ... = n
a'x + b'y + c'z + ... = n'
a"x + b"y + c"z + ... = n"

which was called by him and those of his time the "Problem of the
Virgins." The enumeration is given by the coefficient of x^n.y^n'.z^n" ...
in the expansion of the fraction

1
----------------------------------------------------------------------
(1 - x^a.y^b.z^c...)(1 - x^a'.y^b'.z^c'...)(1 - x^a".y^b".z^c"...) ...

which enumerates the partitions of the multipartite number /nn'n"...
into the parts

/abc..., /a'b'c'..., /a"b"c"..., ...

Sylvester has determined an analytical expression for the coefficient of
x^n in the expansion of

1
------------------------------
(1 - x^a)(1 - x^b)...(1 - x^i)

To explain this we have two lemmas:--

_Lemma 1._--The coefficient of x^-1, i.e., after Cauchy, the residue in
the ascending expansion of (1 - e^x)^-i, is -1. For when i is unity, it
is obviously the case, and

(1 - e^x)^-i-1 = (1 - e^x)^-i + e^x(1 - e^x)^-i-1

d 1
= (1 - e^x)^-i + -- (1 - e^x)^-i.--.
dx i

d 1
Here the residue of -- (1 - e^x)^-i.-- is zero, and therefore the residue
dx i
of (1 - e^x)^-i is unchanged when i is increased by unity, and is
therefore always -1 for all values of i.

_Lemma 2._--The constant term in any proper algebraical fraction
developed in ascending powers of its variable is the same as the
residue, with changed sign, of the sum of the fractions obtained by
substituting in the given fraction, in lieu of the variable, its
exponential multiplied in succession by each of its values (zero
excepted, if there be such), which makes the given fraction infinite.
For write the proper algebraical fraction

c_{[lambda],[mu]} [gamma]_[lambda]
F(x) = [Sigma][Sigma]-------------------- + [Sigma]---------