A274345 -id:A274345 - 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))};

A274346 Denominators 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, 2, 3, 4, 5, 3, 7, 8, 9, 5, 11, 3, 13, 7, 15, 16, 17, 9, 19, 5, 21, 11, 23, 3, 25, 1, 27, 7, 29, 3, 31, 32, 33, 17, 35, 9, 37, 19, 39, 5, 41, 21, 43, 11, 9, 23, 47, 6, 49, 25, 51, 13, 53, 27, 55, 7, 57, 29, 59, 15, 61, 31, 63, 64, 1, 33, 67, 17, 3, 35, 71, 9, 73, 37, 75, 19, 11, 39, 79, 10

Offset: 0

Wolfdieter Lang, Jun 30 2016

The numerators are given in A274345, where also details and the first rationals are given.

The Mathematica program below gives the rationals r(n-1), n = 1..50.

Cf. A227503, A274345.

Table[SeriesCoefficient[Log[EllipticNomeQ[16 x]/x]/8, {x, 0, n}], {n, 1, 50}] // Denominator (* Vaclav Kotesovec, Jun 30 2016 *)

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

(log(q) - log(k^2/16))/(8*k^2/16) = Sum_{n >= 0} (A274345(n)/a(n))*(k^2/16)^n.

More terms from Vaclav Kotesovec, Jun 30 2016

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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A274346 Denominators 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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