After successful identification of the general patterns of the series defined in [1] and [2], the last series expansion that I would like to determine is given by $$ f(x) = 1 + \frac{5}{12} \, x^2 + \frac{35}{128} \, x^4 + \frac{105}{512} \, x^6 + \frac{2695}{16384} \, x^8 + \frac{9009}{65536} \, x^{10} + \frac{495495}{4194304} \, x^{12} + \dots \, . $$
As suggested previously by HallaSurvivor and later by Gary, it is highly recommend to utilize the Online Encyclopedia of Integer Sequences (OEIS) as a valuable resource for identifying common functions based on their series coefficients.
However, in this particular case, the function provided does not yield any relevant information on the OEIS platform.
After extensive exploration, I have been considering the potential utilization of a combination of complete elliptic integrals of the first and second kind, potentially along with their derivatives, as a means to achieve the desired result. Regrettably, my efforts thus far have not yielded any success.
Given the hesitation to pose a third consecutive question of the very same nature, especially considering the sad downvotes received on the two previous linked questions, I am still optimistic that someone within this great community can offer their expertise and provide useful hints that will ultimately aid in solving the problem.
A few more additional terms: $$ f(x) = \dots + \frac{1738165}{16777216} \, x^{14} + \frac{99075405}{1073741824} \, x^{16} + \frac{357271915}{4294967296} \, x^{18} + \frac{31225565371}{412316860416} \, x^{20} + \dots \, . $$
Following the previous answer the solution can be written as $$ f(x) = \frac{1}{x^4} \left[ A(x^2) K(x^2) + B(x^2) E(x^2) \right] , \tag{1} $$ where $K$ and $E$ stand for the complete elliptic integrals of the first and second kind, respectively, and $A$ and $B$ are unknown polynomials.
Expanding $A(x^2)$ and $B(x^2)$ as power series of $x^2$ and identifying the terms given by (1) with those corresponding to the original series, it is easy to show that $$ f(x) = \frac{16}{9\pi x^4} \left[ (x^2+2) K(x^2) - 2(x^2+1) E(x^2) \right]. $$