General pattern for the series $S = x + \frac{1}{4}x^3 + \frac{9}{64} x^5 + \frac{25}{256} x^7+ \dots$

Question

Is there a way to obtain the general term of the following series $$ S = x + \frac{1}{4}x^3 + \frac{9}{64} x^5 + \frac{25}{256} x^7 + \frac{1225}{16384} x^9 + \frac{3969}{65536} x^{11} + \frac{53361}{1048576} x^{13} + \frac{184041}{4194304} x^{15} + \frac{41409225}{4194304} x^{17} + \dots . $$

What I can see is that the denominator takes initially the form of power of 4 but I cannot tell much how the pattern of the numerator evolves. Any thoughts?

Answer 1

When presented with a sequence of integers (like $1,9,25,1225,3969,53361,\ldots$) a fantastic first idea is to consult the Online Encyclopedia of Integer Sequences. In particular, we can search the OEIS for your sequence, and we find it agrees with A038534, the "Numerators of the Coefficients of EllipticK/Pi".

In this post there are references to formulas for these numbers, as well as places in the literature where they have occurred.

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I hope this helps ^_^

Answer 2

The numerators are numerators of the hypergeometric series ${}_{2}F_{1}(1/2,1/2;1;x)$.