OFFSET
0,3
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
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.
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!.
a(n) ~ log(2) * 2^(4*n - 1/2) * n^n / (sqrt(Pi*n) * exp(n)). - Vaclav Kotesovec, Jul 10 2016
EXAMPLE
G.f. = x + 21*x^2 + 740*x^3 + 37310*x^4 + 2460024*x^5 + 200770416*x^6 + 19551774528*x^7 + ...
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 *)
PROG
(PARI) {a(n) = if( n<0, 0, sum( k=1, n, 1 / (2*k - 1) / k) / 4 * (2*n)!^2 / n!^3)};
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Michael Somos, Dec 05 2002
STATUS
approved