There is related information stated in Appendix A 7.56 of the second edition Concrete Mathematics by R. L. Graham, D. E. Knuth and O. Patashnik. Problem 7.56 (p. 380) is one of two research problems in section Exercises of chapter 7. The following is more or less verbatim from the book.
Problem 7.56: Prove that there is no simple closed form for the coefficient of $z^n$ in $\left(1+z+z^2\right)^n$, as a function of $n$, in some large class of simple closed forms.
Appendix A 7.56 provides the following answer:
Answer 7.56: Euler [113] showed that this number is also $[z^n]1/\sqrt{1-2z-3z^2}$, and he gave the formula $t_n=\sum_{k\geq 0}n^{\underline{2k}}/k!^2=\sum_{k}\binom{n}{k}\binom{n-k}{k}$. He also discovered a memorable failure of induction while examining these numbers: Although $3t_n-t_{n+1}$ is equal to $F_{n-1}(F_{n-1}+1)$ for $0\leq n\leq 8$, this empirical law mysteriously breaks down when n is 9 or more! George Andrews [12] has explained this mystery by showing that the sum $\sum_{k}[z^{n+10k}](1+z+z^2)^n$ can be expressed as a closed form in terms of Fibonacci numbers. H. S. Wilf observes that $[z^n](a+bz+cz^2)^n=[z^n]1/f(z)$ , where $f(z)=\sqrt{1-2bz+(b^2-4ac)z^2}$ (see [373, page 159]), and it follows that the coeffcients satisfy
\begin{align*}
(n+1)A_{n+1}-(2n+1)bA_n+n(b^2-4ac)A_{n-1}=0.
\end{align*}
The algorithm of Petkovšek [291] can be used to prove that this recurrence has a closed form solution as a finite sum of hypergeometric terms if and only if $abc(b^2-4ac)=0$. Therefore in particular, the middle trinomial coeffcients have no such closed form. The next step is presumably to extend this result to a larger class of closed forms (including harmonic numbers and/or Stirling numbers, for example).
[12] George E. Andrews, Euler's 'exemplum memorabile inductionis fallacis' and q-trinomial coeffcients, Journal of the American Mathematical Society 3 (1990), 653-669.
[113] L. Eulero, Observationes analyticae, Novi commentarii academiae sci- 299, 575, 636. entiarum imperialis Petropolitanae 11 (1765), 124{143. Reprinted in his Opera Omnia, series 1, volume 15, 50-69.
[291] Marko Petkovšek, Hypergeometric solutions of linear recurrences with polynomial coeffcients, Journal of Symbolic Computation 14 (1992), 243-264.
[373] Herbert S. Wilf, generatingfunctionology. Academic Press, 1990; second edition, 1994.
Note: I looked in [373] Herbert S. Wilf, generatingfunctionology, second edition, but unfortunately I could not find the cited information. I could find there some related information in chapter 5, Exercises 2 - 4. For instance exercise 4 (a):
- 4 (a): Define, for all $n\geq 0, \gamma_n=[x^n](1+x+x^2)^n$. Use the result of exercise 3 above to prove that for $n\geq 0$,
\begin{align*}
\gamma_n=[x^n]\left\{\frac{1}{\sqrt{1-2x-3x^2}}\right\}.
\end{align*}
