# [seqfan] Re: A132091 same sequence?

Max Alekseyev maxale at gmail.com
Sun Jun 7 03:44:09 CEST 2009

```G.f. of the number of integer partitions with each part appearing at
least two times is:

f(x) = \prod_{k>=1} (1 + x^(2k) + x^(3*k) + ...)
= \prod_{k>=1} (1/(1-x^k) - x^k)
= \prod_{k>=1} (1 - x^k + x^(2*k)) / (1 - x^k)

Excluding parts divisible by 3, we have:

f(x) / f(x^3) = \prod_{k>=1} (1 - x^k + x^(2*k)) * (1 - x^(3*k)) / (1
- x^k) / (1 - x^(3*k) + x^(6*k))
= \prod_{k>=1} (1 - x^k + x^(2*k)) * (1 + x^k + x^(2*k)) / (1 -
x^(3*k) + x^(6*k))
=  \prod_{k>=1} (1 + x^(2*k) + x^(4*k)) / (1 - x^(3*k) + x^(6*k))

that matches the g.f. given in  A132091:

%F A132091 G.f.: Product_{k>0} (1+x^(2*k)+x^(4*k))/ (1-x^(3*k)+x^(6*k)).

Regards,
Max

On Sat, Jun 6, 2009 at 6:50 PM,  <rhhardin at att.net> wrote:
> The number of partitions of n into parts not divisible by 3 with every part repeated at least twice seems to be the same as A132091
>
> Expansion of psi(q^3)* chi(-q^9)/ f(-q^2) in powers of q where psi(), chi(), f() are Ramanujan theta functions.
>
> which I have zero intuition for.
>
> A b-file for the partition into n parts etc. is at http://home.att.net/~rhhardin/b132091.txt
> n=1..902, if somebody wants to check higher terms on it.
> --
> rhhardin at mindspring.com
> rhhardin at att.net (either)
>
>
>
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>

```