The sequence is:
1, 2, 3, 5, 8, 12, 18, 25, 35, 50, 69, 93, 126, 167, 220, 290, 377, 486, 627, 800, 1017, 1290, 1623, 2032, 2542, 3161, 3917, 4843,...
It is related to partitions of $n$. It is a subsequence of a sequence that has the generating function:
$$G(t,x) = \prod_{i>=1} \dfrac{1 - x^{prime(i)}}{(1 - x^i)\cdot(1 - tx^{prime(i)})}$$
That triangle has row lengths 1,1,2,2,3,3,4,4,... If we start with row and column indices of zero, my sequence is formed by taking the $n$th element from the $3n$th row in that triangle.
I don't know what other information might be necessary to answer the question, so ask if needed.
EDIT:
My sequence in now https://oeis.org/A299731. If the answer to my question is in the affirmative the generating function can be added to A299731, or I will add it and link to this question, so that the answerer gets the credit.