[seqfan] Re: Are the partials sums of A025147 equal to A038348?

[William Keith]

On Apr 1, 2010, at 9:18 PM, Jonathan Post wrote:

> Are the partials sums of A025147 Number of partitions of n into
> distinct parts >= 2.
> equal to A038348  Expansion of (1/(1-x^2))*Product((1/(1-x^(2m+1)),
> m=0..inf. (equivalently Number of partitions of n with at most one
> even part)?

Yes.  The g.f. for partitions of n into distinct parts >= 2 is Product(1+x^m), m >= 2.  The g.f. for its partial sums is then

(1+x+x^2+x^3+...) Product(1+x^m), m >= 2
= 1/(1-x) Product(1+x^m), m >= 2.

Transform thus:

1/(1-x) Product(1+x^m), m >= 2
= 1/(1-x) Product((1-x^(2m))/(1-x^m)) , m>= 2   (multiply by Product((1-x^m)/(1-x^m)), m>=2)
= 1/(1-x^2) Product(1-x^(2m+1), m >= 0  (pull out 1/(1-x^2), push in 1/(1-x), cancel all even powers >= 4 in denominator)

There is probably a straightforward bijective proof as well.

Cordially,
William Keith