Partitions into divisors... A018818

[FRANCISCO SALINAS]

Gordon Royle wrote:

>I am interested in alternative formulations for the number of partitions of 
>an integer n into divisors of n...
>
>The sequence is in OEIS as A018818,
>
>http://www.research.att.com/projects/OEIS?Anum=A018818
>
>and the comments observe that the value a(n) is the coefficient of x^n in 
>1/(Product(1-x^d)) where d ranges over all divisors of n.
>
>The problem with this expression is that the "generating function" is 
>different for each value of n and we only pull one coefficient out of it..
>
>Does anyone know if this sequence has been studied

Not exactly what you look for, but this reference from MathScinet ( 
http://www.ams.org/mathscinet-getitem?mr=0469864 ) goes for the comments:

Gupta, Hansraj. Partitions of $n$ into divisors of $m$. Indian J. Pure Appl. 
Math. 6 (1975), no. 11, 1276--1286. MR0469864 (57 #9645)

Review by M. S. Cheema:

Let $P_m(n)$ denote the number of partitions of $n$ into the divisors of 
$m$. Simple manipulation shows that the generating fucntion $$ \align 
\sum{}_{n=0}^\infty P_m(n)x^n & =\prod{}_{d|m}(1-x^d)^{-1}\\ & 
=(1-x^m)^{-1}\prod{}_{d|m}(1+x^d+x^{2d}+\cdots+x^{m-d}), \endalign $$ and 
$P_m(n)=\sum_{k=0}^{t-1}C(r+km)\left( \matrix q+t-k-1 \\ t-1 \endmatrix 
\right)$, where $\sum_{a=0}^uC(a)x^a=\prod_{d|m}(1+x^d+\cdots+x^{m-d})$, 
$u=mt-\sigma$. The $c$'s can be easily evaluated directly or by a simple 
algorithm involving a suitable sequence of matrix multiplications.

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