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How to express this series in closed form? $\sum_{i=1}^{\infty}\frac{(3i)!}{(i!)^3}x^{i}$

Motive of the generating function is to evaluate the number of the paths from the $(0,0,0)$ to $(n,n,n)$ not passing through $(i,i,i)$ ($1\leq i\leq n-1)$. The answer is coefficient of $x^n$ in $\sum_{k=1}^{n}(-1)^{k-1}\left(\sum_{i=1}^{n}\frac{(3i)!}{(i!)^3}x^{i}\right)^{k}.$

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    see De Bruijn's S(3,n) at [OEIS](https://oeis.org/A006480). You should find Shaktal's answer for example !2012-08-28

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The summand $c_n = x^n (3n)!/(n!)^3$ is a hypergeometric term, meaning that $c_{n+1}/c_n$ is a rational function of $n$. Indeed: $ \frac{c_{n+1}}{c_n} = \frac{x^{n+1}}{x_n} \frac{(3n+3)!}{(3n)!} \left(\frac{n!}{(n+1)!}\right)^3 = x \frac{(3n+3)(3n+2)(3n+1)}{(n+1)^3} = 27 x \frac{\left(n+\frac{1}{3}\right)}{(n+1)} \frac{\left(n+\frac{2}{3}\right)}{(n+1)} $ Such a recurrence equation implies that $ c_n = c_0 \prod_{k=0}^{n-1} 27 x \frac{\left(k+\frac{1}{3}\right)}{(k+1)} \frac{\left(k+\frac{2}{3}\right)}{(k+1)} = c_0 \frac{(27 x)^n}{n!} \frac{\left(\frac{1}{3}\right)_n \left(\frac{2}{3}\right)_n}{(1)_n} $ Now the sum in question get's written as a defining series of the Gauss hypergeometric function: $ \sum_{i=0}^\infty \binom{3i}{i,i,i} x^i = \sum_{n=0}^\infty \frac{(27 x)^n}{n!} \frac{\left(\frac{1}{3}\right)_n \left(\frac{2}{3}\right)_n}{(1)_n} = {}_2F_1\left(\frac{1}{3}, \frac{2}{3}; 1; 27 x\right) $ The result for summation from $i=1$ is obtained by subtracting the first term.

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Here is a closed form by Maple in terms of the hypergeometric function.

$ 6 \,x \,{ _3F_{2}\left(1,\frac{4}{3},\frac{5}{3};\,2,2;\,27\,x\right)}\,. $