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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.

%I A227503 #19 Oct 06 2019 06:16:36
%S A227503 1,13,184,2701,40456,613720,9391936,144644749,2238445480,34772271208,
%T A227503 541801226176,8463116730712,132472258939840,2077232829015616,
%U A227503 32621327116946944,512963507737401997,8075477240446327528,127258797512376887176,2007225253307641799872
%N 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.
%C A227503 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 + ... .
%C A227503 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
%H A227503 Vaclav Kotesovec, <a href="/A227503/b227503.txt">Table of n, a(n) for n = 1..800</a>
%H A227503 A. Kneser, <a href="https://eudml.org/doc/149645">Neue Untersuchung einer Reihe aus der Theorie der elliptischen Funktionen</a>, J. reine u. angew. Math. 157, 1927, 209 - 218.
%H A227503 R. Fricke, <a href="http://dx.doi.org/10.1007/978-3-642-20954-3_1">Die elliptischen Funktionen und ihre Anwendungen</a>, Dritter Teil, Springer-Verlag, 2012., p. 2, eq. (5).
%e A227503 G.f. = x + 13*x^2 + 184*x^3 + 2701*x^4 + 40456*x^5 + 613720*x^6 + 9391936*x^7 + ...
%t A227503 a[ n_] := If[ n < 0, 0, n SeriesCoefficient[ Log[ EllipticNomeQ[ 16 x] / x] / 8, {x, 0, n}]];
%o A227503 (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))};
%Y A227503 Cf. A002103, A274345, A274346.
%K A227503 nonn
%O A227503 1,2
%A A227503 _Michael Somos_, Jul 13 2013