Is there a database for sequences indexed by partitions similar to Sloane's OEIS? I mean, I am aware that in the OEIS there are some arrays indexed by partitions, but I feel as though most of such sequences that frequently appear in combinatorial literature are not there.

One example of a sequence I'd really like to recognise begins as follows:

$a[1]=1$,

$a[2]=a[1^2]=2$,

$a[3]=a[1^3]=4$, $a[2,1]=8$,

$a[4]=a[1^4]=8$, $a[3,1]=a[2,1^2]=16$, $a[2^2]=24$,

$a[5]=a[1^5]=16$, $a[4,1]=a[2,1^3]=32$, $a[3,2]=a[2^2,1]=52$, $a[3,1^2]=48$.

The obvious patterns $a[\lambda]=a[\lambda^t]$ and $a[n]=2^{n-1}$, $a[n-1,1]=2^n$ do hold in general, if it helps.