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I think it is a bug.

A workaround for now is to use coeffs instead of coefficients.

Here is what you get then.

sage: for n in (0..4):
....:     print g(x,n).series(x,3*n).coeffs()
....:     
/opt/s/sage-7.4/src/bin/sage-ipython:2: DeprecationWarning: coeffs is deprecated. Please use coefficients instead.
See http://trac.sagemath.org/17438 for details.
  # -*- coding: utf-8 -*-
[[1, 0]]
[[1, 0], [1, 1], [1, 2]]
[[1, 0], [2, 1], [2, 2], [2, 3], [1, 4]]
[[1, 0], [3, 1], [3, 2], [6, 3], [3, 4], [3, 5], [1, 6]]
[[1, 0], [4, 1], [4, 2], [12, 3], [6, 4], [12, 5], [4, 6], [4, 7], [1, 8]]

Note however that coeffs is deprecated, so it might disappear in a future version of Sage.

I'll edit later today with more on the bug.