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Coefficients in the expansion of q^(1/2) in odd powers of k/4, where q is the Jacobi nome and k^2 the parameter of elliptic functions. Also coefficients in the expansion of q in odd powers of (1/4)*(1 - k') / (1 + k') with k'^2 the complementary parameter.
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%I #16 Oct 07 2019 05:43:28

%S 1,4,34,360,4239,53148,694582,9348664,128625067,1800131564,

%T 25538105486,366348201176,5304067812296,77394671803040,

%U 1136872705730600,16796605751564320,249415741237963837

%N Coefficients in the expansion of q^(1/2) in odd powers of k/4, where q is the Jacobi nome and k^2 the parameter of elliptic functions. Also coefficients in the expansion of q in odd powers of (1/4)*(1 - k') / (1 + k') with k'^2 the complementary parameter.

%C k' is the square root of the complementary parameter of elliptic functions. In the Abramowitz-Stegun (A-St) reference, p. 569, k'^2 is called m_1. The relation between k'^2 and k^2, the parameter (called m in A-St), is k'^2 = 1 - k^2.

%C The expansion of q in odd powers of (1/4)*(1 - k') / (1 + k') appears in the Kneser reference, p. 218, where it is attributed to L. Lindelöf. It is obtained from the expansion of sqrt(q) in odd powers of k/4, namely q^{1/2} = Sum_{n >= 0} a(n)*(k/4)^(2*n+1), which results from the expansion -Pi*K'/K = log(q) = log(k^2/16) + log(1 + Sum_{n>=1} A005797(n+1)*(k^2/16)^n) = log(k^2/16) + 8*(k^2/16) + 52*(k^2/16)^2 + ... (see A-St, p. 591, 17. 3.21, Kneser, p. 216, Fricke, eq. (4), p. 2, and A227505, A274345/A274346). The fact that a replacement of q by q^2 means a replacement of k by (1 - k')/(1 + k') is used (Landen transformation).

%H Vaclav Kotesovec, <a href="/A274344/b274344.txt">Table of n, a(n) for n = 1..800</a>

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy], pp. 569, 591.

%H R. Fricke, <a href="http://dx.doi.org/10.1007/978-3-642-20954-3_1">Die elliptischen Funktionen und ihre Anwendungen</a>, Dritter Teil, Springer-Verlag, 2012., p. 2, eq. (4).

%H A. Kneser, <a href="https://eudml.org/doc/149645">Neue Untersuchung einer Reihe aus der Theorie der elliptischen Funktionen</a>, J. reine u. angew. Math. 157, 1927, 209 - 218.

%F q^{1/2} = Sum_{n >= 0} a(n)*(k/4)^(2*n+1).

%F q = Sum_{n >= 0} a(n)*((1/4)*(1 - k')/(1 + k'))^(2*n+1).

%t CoefficientList[Series[Sqrt[EllipticNomeQ[16*x]/x], {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 07 2019 *)

%Y Cf. A005797, A227505, A274345, A274346.

%K nonn,easy

%O 1,2

%A _Wolfdieter Lang_, Jun 30 2016