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sl^(-1)(x) := int Integral_{s=0..x} 1/sqrt(1-s^4)) ds = x + x^5/10 + x^9/24 + 5*x^13/208 + ....
The coefficients in this expansion can be calculated using nested derivatives as follows (see [Dominici, Theorem 4.1]): Let f(x) = sqrt(1-x^4). Define the nested derivative D^n[f](x) by means of the recursion
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S. Binski, and T. R. Hagedorn, <a href="http://www.tcnj.edu/~hagedorn/papers/CapstonePapers/Binski/CapstoneBinskiLemniscate
D. Diego Dominici, <a href="http://arxiv.org/abs/math/0501052">Nested derivatives: A simple method for computing series expansions of inverse functions</a>, arXiv:math/0501052v2 [math.CA], 2005.
Markus Kuba, and Alois Panholzer, <a href="http://arxiv.org/abs/1411.4587">Combinatorial families of multilabelled increasing trees and hook-length formulas</a>, arXiv:1411.4587 [math.CO], 2014.
Erik Vigren, and Andreas Dieckmann, <a href="https://doi.org/10.3390/sym12061040">Simple Solutions of Lattice Sums for Electric Fields Due to Infinitely Many Parallel Line Charges</a>, Symmetry (2020) Vol. 12, No. 6, 1040.
Zachary P. Bradshaw and Christophe Vignat, <a href="https://arxiv.org/abs/2407.02365">Berndt-type Integrals: Unveiling Connections with Barnes Zeta and Jacobi Elliptic Functions</a>, arXiv:2407.02365 [math.CA], 2024. See p. 9.
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a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ JacobiSD[ x, 1/2] 2^((n - 1)/2), {x, 0, n}]]; (* Michael Somos, Jan 17 2017 *)
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ JacobiSN[x, -1], {x, 0, n}]]; (* Michael Somos, May 26 2021 *)
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