login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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. 3
1, 13, 184, 2701, 40456, 613720, 9391936, 144644749, 2238445480, 34772271208, 541801226176, 8463116730712, 132472258939840, 2077232829015616, 32621327116946944, 512963507737401997, 8075477240446327528, 127258797512376887176, 2007225253307641799872 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
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
LINKS
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).
EXAMPLE
G.f. = x + 13*x^2 + 184*x^3 + 2701*x^4 + 40456*x^5 + 613720*x^6 + 9391936*x^7 + ...
MATHEMATICA
a[ n_] := If[ n < 0, 0, n SeriesCoefficient[ Log[ EllipticNomeQ[ 16 x] / x] / 8, {x, 0, n}]];
PROG
(PARI) {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))};
CROSSREFS
Sequence in context: A297581 A268413 A274345 * A091540 A057799 A057801
KEYWORD
nonn
AUTHOR
Michael Somos, Jul 13 2013
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified February 28 01:44 EST 2024. Contains 370379 sequences. (Running on oeis4.)