A078791 Expansion of Auxiliary function L(1-m) / 4 in powers of m / 16.

0, 1, 21, 740, 37310, 2460024, 200770416, 19551774528, 2213488134000, 285711909912000, 41419784380740480, 6663725042739448320, 1178209566488368028160, 227096910697908706560000

Offset: 0

Author

Michael Somos, Dec 05 2002

Keywords

• nonn
• easy

Comments

Nome q(m) = x exp(8 * (Sum_{n>0} a(n) * x^n / n!) / (Sum_{n>=0} binomial(2n, n)^2 * x^n)) where x = m / 16.

The Fricke reference on page 2 has equation "(3) Pi i omega = -Pi K'/K = log k^2 - 4 log 2 + F_1(1/2, 1/2; k^2) / F(1/2, 1/2, 1; k^2), wo F_1 und F ..." where F_1 = 8 * Sum_{n>0} a(n) * x^n / n! with x = m / 16 = (k / 4)^2. - Michael Somos, Jul 14 2013

Examples

			G.f. = x + 21*x^2 + 740*x^3 + 37310*x^4 + 2460024*x^5 + 200770416*x^6 + 19551774528*x^7 + ...
		

References

• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 591.
• J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 9.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..300
• 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].
• R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Dritter Teil, Springer-Verlag, 2012.

Crossrefs

Cf. A000984, A005797, A098118.

Programs

Mathematica

a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ Log[ EllipticNomeQ[ 16 x] / x] Hypergeometric2F1[ 1/2, 1/2, 1, 16 x] / 8, {x, 0, n}]]; (* Michael Somos, Jul 14 2013 *)
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ Log[ EllipticNomeQ[ 16 x] / x] EllipticK[ 16 x] / (4 Pi), {x, 0, n}]]; (* Michael Somos, Jul 14 2013 *)
a[ n_] := If[ n < 0, 0, n! Binomial[ 2 n, n]^2 Sum[ 1/k, {k, n + 1, 2 n}] / 2]; (* Michael Somos, Jul 14 2013 *)
a[ n_] := If[ n < 0, 0, n! Binomial[ 2 n, n]^2 (HarmonicNumber[2 n] - HarmonicNumber[n]) / 2]; (* Michael Somos, Apr 14 2015 *)

PARI

{a(n) = if( n<0, 0, sum( k=1, n, 1 / (2*k - 1) / k) / 4 * (2*n)!^2 / n!^3)};

Formula

E.g.f.: L(1-m) = log(16 / m) (K(m) / Pi) - K(1-m) = 4 Sum_{n>0} a(n) (m/16)^n / n!.

2 * a(n) = A098118(n) * A000984(n). - Michael Somos, Apr 14 2015

a(n) ~ log(2) * 2^(4*n - 1/2) * n^n / (sqrt(Pi*n) * exp(n)). - Vaclav Kotesovec, Jul 10 2016