This is the number of terms from the first to the -power of x⁹- element: 1, 1, 2, 3, 5, 7, 11, 15, 22, 30

I threw the sequence to The On-Line Encyclopedia of Integer Sequences

http://oeis.org/search?q=1%2C2%2C3%2C5%2...&go=Search

and I got many different ways to calculate it. The first one is: "number of partitions of n (the partition numbers)."

which PariGP can calculate with the numbpart() function:

gp > a=vector(20,k,numbpart(k))

%3 = [1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627]

it works up to 56, but I truncated the series, so it should not work beyond that.

For example, the 4th element, for x⁴, has 5 elements, because 4 can be partitioned in 5 different ways.

For each partition, there is an index i for each number in that partition:

I have the result, but I do not yet know how to get it.