A274346 -id:A274346 - cp's OEIS frontend

A227503 q = x * exp( 8 * (Sum_{k>0} a(k) * x^k / k)) where x = m/16, q is the elliptic nome and m = k^2 is the parameter.

1, 13, 184, 2701, 40456, 613720, 9391936, 144644749, 2238445480, 34772271208, 541801226176, 8463116730712, 132472258939840, 2077232829015616, 32621327116946944, 512963507737401997, 8075477240446327528, 127258797512376887176, 2007225253307641799872

Offset: 1

Michael Somos, Jul 13 2013

nonn

The Fricke reference has equation Pi i omega / 4 = log (sqrt(k) / 2) + 2 (sqrt(k) / 2)^4 + 13 (sqrt(k) / 2)^8 + 368/3 (sqrt(k) / 2)^12 + 2701/2 (sqrt(k) / 2)^16 + ... .

This can be written (with Pi i omega / 4 = log(q)/4) as (log(q) - log(k^2/16)) / (8*k^2/16) = Sum_{n >= 0} (a(n+1)/(n+1))*(k^2/16)^n. See also the Kneser reference, p. 216. Note that the rational coefficients a(n+1)/(n+1) are not reduced to lowest terms. For the reduced rational coefficients see A274345 / A274346. - Wolfdieter Lang, Jun 30 2016

			G.f. = x + 13*x^2 + 184*x^3 + 2701*x^4 + 40456*x^5 + 613720*x^6 + 9391936*x^7 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 1..800
A. Kneser, Neue Untersuchung einer Reihe aus der Theorie der elliptischen Funktionen, J. reine u. angew. Math. 157, 1927, 209 - 218.
R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Dritter Teil, Springer-Verlag, 2012., p. 2, eq. (5).

Cf. A002103, A274345, A274346.

a[ n_] := If[ n < 0, 0, n SeriesCoefficient[ Log[ EllipticNomeQ[ 16 x] / x] / 8, {x, 0, n}]];

{a(n) = local(A); if( n<1, 0, A = x * O(x^n); n * polcoeff( log( serreverse( x * (eta(x + A) * eta(x^4 + A)^2 / eta(x^2 + A)^3)^8 ) / x) / 8, n))};

A274345 Numerators of coefficients in the expansion of (log(q) - log(k^2/16))/(8*k^2/16) in powers of k^2/16, where q is the Jacobi nome and k^2 the parameter of elliptic functions.

Original entry on oeis.org

1, 13, 184, 2701, 40456, 306860, 9391936, 144644749, 2238445480, 17386135604, 541801226176, 2115779182678, 132472258939840, 1038616414507808, 32621327116946944, 512963507737401997, 8075477240446327528, 63629398756188443588, 2007225253307641799872, 7921211894405933627674, 500517296244244008379456

Offset: 0

Wolfdieter Lang, Jun 30 2016

For the denominators see A274346.

The rationals r(n) = a(n)/A274346(n) are given by A227503(n+1)/(n+1) reduced to lowest terms. See A227503 for details, references and links.

			The first rationals r(n) = a(n)/A274346(n) are: 1/1, 13/2, 184/3, 2701/4, 40456/5, 306860/3, 9391936/7, 144644749/8, 2238445480/9, 17386135604/5, 541801226176/11, 2115779182678/3, 132472258939840/13, 1038616414507808/7, 32621327116946944/15, ...

Cf. A227503, A274346.

(* See the program for r(n-1), n >= 1, in A274346. *)

a(n) = numerator(A227503(n+1)/(n+1)), n >= 0.

(log(q) - log(k^2/16))/(8*k^2/16) = Sum_{n >= 0} (a(n)/A274346(n))*(k^2/16)^n.

A274344 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.

Original entry on oeis.org

1, 4, 34, 360, 4239, 53148, 694582, 9348664, 128625067, 1800131564, 25538105486, 366348201176, 5304067812296, 77394671803040, 1136872705730600, 16796605751564320, 249415741237963837

Offset: 1

Wolfdieter Lang, Jun 30 2016

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.

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).

Vaclav Kotesovec, Table of n, a(n) for n = 1..800
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy], pp. 569, 591.
R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Dritter Teil, Springer-Verlag, 2012., p. 2, eq. (4).
A. Kneser, Neue Untersuchung einer Reihe aus der Theorie der elliptischen Funktionen, J. reine u. angew. Math. 157, 1927, 209 - 218.

Cf. A005797, A227505, A274345, A274346.

CoefficientList[Series[Sqrt[EllipticNomeQ[16*x]/x], {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 07 2019 *)

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

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

cp's OEIS Frontend

A227503 q = x * exp( 8 * (Sum_{k>0} a(k) * x^k / k)) where x = m/16, q is the elliptic nome and m = k^2 is the parameter.

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A274345 Numerators of coefficients in the expansion of (log(q) - log(k^2/16))/(8*k^2/16) in powers of k^2/16, where q is the Jacobi nome and k^2 the parameter of elliptic functions.

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A274344 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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