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A078791 Expansion of Auxiliary function L(1-m) / 4 in powers of m / 16. 2
0, 1, 21, 740, 37310, 2460024, 200770416, 19551774528, 2213488134000, 285711909912000, 41419784380740480, 6663725042739448320, 1178209566488368028160, 227096910697908706560000 (list; graph; refs; listen; history; text; internal format)
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!.

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

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

Cf. A000984, A005797, A098118.

Sequence in context: A297504 A250059 A250060 * A201069 A143002 A062755

Adjacent sequences:  A078788 A078789 A078790 * A078792 A078793 A078794

KEYWORD

nonn,easy

AUTHOR

Michael Somos, Dec 05 2002

STATUS

approved

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Last modified August 20 18:56 EDT 2019. Contains 326154 sequences. (Running on oeis4.)