I was playing with the idea of a sequence transform such that if

b=Tr(a)

then b(n) is the number of ways to partition n into at most a(1) copies of 1,
a(2) copies of 2, etc.

This is easy to represent as a generating function as

B(x) = prod( (1-x^(i*(a(i)+1)))/(1-x^i), i=1..infinity ).

Applied to the all ones sequence it of course gives A000009 partitions into
distinct parts (at most one copy of each part.)

To the all 2's sequence A000726

%I A000726 M0316 N0116
%S A000726 1,1,2,2,4,5,7,9,13,16,22,27,36,44,57,70,89,108,135,163,202,243,297,355,
%T A000726 431,513,617,731,874,1031,1225,1439,1701,1991,2341,2731,3197,3717,4333,
%U A000726 5022,5834,6741,7803,8991,10375,11923,13716,15723,18038,20628
%N A000726 Partitions of n into parts prime to 3.
%C A000726 Also partitions with at most 2 parts of size 1 and all differences between parts at distance 3 are greater than 1.

3's A001935, 4's A035959

In fact the number of partitions into at most k copies of each part is the
number of partitions into parts not divisible by k+1.

The interesting part is those sequences not in the EIS (yet).

Apply the transform to a(n)=n and get (starting at 0)
1 1 1 2 3 4 5 7 10 13 17 22 28 36 46 58 73 91 114...
partitions into at most 1 copy of 1, 2 copies of 2, 3 copies of 3,...
is also partitions into non pronic (n*(n+1) for some n) numbers
Soon to be A052335

This sequence that shifts left under the transform
0 1 1 1 2 2 3 4 5 7 9 12 15 18 23 29 35 44 54 66 81 98...
(Soon to be A052336)
applying the transform gives
1 1 1 2 2 3 4 5 7 9 12 15 18 23 29 35 44 54 66 81 98 119...

Like most left shifting eigensequences this can be described as a count of
rooted trees (but in this case it's a complicated description).

There is a 13 term match with A039853 btw.

A more interesting feature:
Applying it to all 1's gives A000009, applying it again gives a larger sequence.

"Virtually" applying it to the "all infinity" sequence gives A000041
(unrestricted) partitions.
Applying it again gives a smaller sequence.

There is a sequence in the middle that they converge to. That sequence begins:

1 1 1 2 2 3 5 6 8 10 13 17 21 27 34 42 53 65 80 98 119...
(Soon to be A052337)

It almost seems like this sequence should be important in some other way.

I deliberately avoided considering the case of partitioning n into zeroes,
avoiding it makes the equations easier to deal with, but it's actually not
necessary, since as long as there are finitely many zeroes allowed in the
partition, there are finitely many partitions.

Christian