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The squares of the Catalan numbers: 1, 1, 4, 25, 196, 1764... are given in OEIS A001246. In the OEIS entry two ordinary generating functions for the series are given in terms of elliptic integrals/hypergeometric functions:

$$-1/4x+(16x-1)\textrm{EllipticK}(4\sqrt{x})/2x\pi + \textrm{EllipticE}(4\sqrt{x})/\pi x,$$ and $${}_3F_2( (1, 1/2, 1/2); (2, 2); 16x) = (-1 + {}_2F_1( (-1/2, -1/2); (1); 16x))/(4x). $$

Unfortunately I cannot find a derivation for either of these expressions. The squared Catalan numbers are given by the Hadamard product of the Catalan number generating funcion with itself, so one approach would be to use the integral form of the Hadamard product, but I am not sure how to evaluate this integral, and Mathematica also struggles (even though it would seems the first expression was probably derived using Mathematica given the notation?)

Were these expressions derived by evaluating such an integral, or perhaps appealing to the recursion relations satisfied by the Catalan numbers like the proof for the g.f. for the Catalan numbers themselves?

maxwelldecoherence
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    Squared Catalan numbers count systems of meanders. Lando and Zvonkin in Meanders derive an integral expression for their generating function on p. 130, although it is not one of OEIS's two expressions, but may be it'll help you get there. – Conifold Sep 22 '21 at 06:12
  • @Conifold This is extremely helpful. Plugging Lando and Zvonkin's integral expression [that they basically obtain using the integral form of Hadamard product/Parseval's theorem] into Mathematica gives an expression in terms of elliptic integrals which I believe with the use of the correct elliptic integral identities is indeed equivalent to the first expression in OEIS. – maxwelldecoherence Sep 22 '21 at 23:40

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