Explicit examples of formal power series which is not rational functions? This MO question https://mathoverflow.net/questions/249541/formal-power-series-is-taylor-expansion-of-rational-function-iff-hankel-determin states that if $k$ is a field and $k[[T]]$ is the power series ring over $k$ then $u(T)\in k[[T]]$ is the power series of some rational function if and only if a condition on Hankel determinant holds. However, I could not think of any explicit power series which does not satisfy this condition.
I know that if $k=\mathbb{R},\mathbb{C}$, there are a lot of examples of power series which are not rational functions. But are there any explicit examples of power series which work for an arbitrary field $k$ ?

A good class of examples is algebraic power series that are not rational. For example, consider the Catalan number generating function
$$c(x) = \frac{1-\sqrt{1-4x}}{2x}= \sum_{n=0}^\infty \frac{1}{n+1}\binom{2n}{n}x^n$$
which is defined over every field. It satisfies $c(x) = 1+xc(x)^2$, and it's not hard to show from this that $c(x)$ is not rational.
