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A227503
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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.
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3
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1, 13, 184, 2701, 40456, 613720, 9391936, 144644749, 2238445480, 34772271208, 541801226176, 8463116730712, 132472258939840, 2077232829015616, 32621327116946944, 512963507737401997, 8075477240446327528, 127258797512376887176, 2007225253307641799872
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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EXAMPLE
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G.f. = x + 13*x^2 + 184*x^3 + 2701*x^4 + 40456*x^5 + 613720*x^6 + 9391936*x^7 + ...
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MATHEMATICA
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a[ n_] := If[ n < 0, 0, n SeriesCoefficient[ Log[ EllipticNomeQ[ 16 x] / x] / 8, {x, 0, n}]];
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PROG
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(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))};
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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