Partition numbers as the number of terms in a expansion

Clear[a, f, s, n, nn];
nn = 20;
a = CoefficientList[Series[1/f[x], {x, 0, nn}], x];
Table[Length[Expand[a[[n]]]], {n, 1, Length[a]}]

{2, 3, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, \
297, 385, 490, 627}

https://mathoverflow.net/questions/374643/is-there-a-connection-between-the-sequence-of-a-finite-number-of-stieltjes-const