Is it possible to compute the following power series, in closed form? (where $x$ is sufficiently small)
$$f(x)=\sum_{d=1}^\infty\frac{(3d)!}{(d)!^3}x^d.$$
I would appreciate any ideas.
2026-03-27 01:42:12.1774575732
power series with multinomial coefficient
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The coefficient sequence is OEIS A006480 with $a(0)=1$. I suggest adding that term to your summation. As another user commented, it is the power series expansion of $\; _2F_1\left(\frac{1}{3},\frac{2}{3};1;27 x\right)$ and is given in the OEIS entry. I regard this as a closed form and unlikely to have a simpler form.