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A186690
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Expansion of - (1/8) theta_3''(0, q) / theta_3(0, q) in powers of q.
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26
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1, -2, 4, -4, 6, -8, 8, -8, 13, -12, 12, -16, 14, -16, 24, -16, 18, -26, 20, -24, 32, -24, 24, -32, 31, -28, 40, -32, 30, -48, 32, -32, 48, -36, 48, -52, 38, -40, 56, -48, 42, -64, 44, -48, 78, -48, 48, -64, 57, -62, 72, -56, 54, -80, 72, -64, 80, -60, 60, -96, 62, -64
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OFFSET
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1,2
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COMMENTS
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If A(x) is the generating function then 1 / Pi = 8 A( exp( -Pi) ). [Plouffe, equation 1.2]
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REFERENCES
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A. P. Prudnikov, Yu. A. Brychkov and O.I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", Chapter 4: "Finite Sums", New York, Gordon and Breach Science Publishers, 1986-1992, Equation (5.1.29.8).
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LINKS
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FORMULA
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Multiplicative with a(2^e) = -(2^e) if e>0, a(p^e) = (p^(e+1) - 1) / (p - 1) if p > 2.
Expansion of (E - (1 - k^2) * K) * K / (2 Pi^2) in powers of the nome q where K, E are complete elliptic integrals.
Expansion of (1/2) x (d phi(x) / dx) / phi(x) in powers of x where phi() is a Ramanujan theta function.
G.f.: Sum_{k>0} - (-1)^k * k * x^k / (1 - x^(2*k)) = Sum_{k>0} x^(2*k-1) / (1 + x^(2*k-1))^2 = (Sum_{k>0} n^2 x^(n^2)) / (Sum_k x^(n^2)).
Dirichlet g.f. zeta(s) *zeta(s-1) *(1-7*2^(-s)+14*4^(-s)-8^(1-s)) / (1-2^(1-s)). - R. J. Mathar, Jun 01 2011
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EXAMPLE
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G.f. = q - 2*q^2 + 4*q^3 - 4*q^4 + 6*q^5 - 8*q^6 + 8*q^7 - 8*q^8 + 13*q^9 + ...
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MATHEMATICA
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a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (1/8) (EllipticE[m] - (1 - m) EllipticK[m]) EllipticK[m]/(Pi/2)^2, {q, 0, n}]];
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PROG
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(PARI) {a(n) = if( n<1, 0, -(-1)^n * sumdiv( n, d, d / gcd(d, 2)))};
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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