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A115977
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Expansion of elliptic modular function lambda in powers of the nome q.
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4
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16, -128, 704, -3072, 11488, -38400, 117632, -335872, 904784, -2320128, 5702208, -13504512, 30952544, -68901888, 149403264, -316342272, 655445792, -1331327616, 2655115712, -5206288384, 10049485312, -19115905536, 35867019904, -66437873664
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OFFSET
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1,1
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COMMENTS
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Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
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REFERENCES
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 591.
J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 121.
A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, p. 23, eq. (37).
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LINKS
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Table of n, a(n) for n=1..24.
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].
M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Eric Weisstein's World of Mathematics, Elliptic Lambda Function
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FORMULA
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Expansion of Jacobi elliptic m = k^2 = (theta_2(q) / theta_3(q))^4 in powers of the nome q.
Expansion of 16 * q * (psi(q^2) / phi(q))^4 = 16 * q * (psi(q^2) / psi(q))^8 = 16 * q * (psi(q) / phi(q))^8 = 16 * q * (psi(-q) / phi(-q^2))^8 = 16 * q / (chi(q) * chi(-q^2))^8 = 16 * q * (f(-q^4) / f(q))^8 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.
Expansion of 16 * (eta(q) * eta(q^4)^2 / eta(q^2)^3)^8 in powers of q.
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 * (1 - v)^2 - 16 * v * (1 - u).
lambda( -1 / tau ) = 1 - lambda( tau ) (see A128692).
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = g(t) where q = exp(2 pi i t) and g() is g.f. for A128692.
G.f.: 16 * q * (Product_{k>0} (1 + q^(2*k)) / (1 + q^(2*k - 1)))^8.
a(n) = 16 * A005798(n). a(n) = -(-1)^n * A014972(n) unless n=0.
Empirical : sum(exp(-2*Pi)^n*a(n),n=1..infinity) = 17-12*sqrt(2). Simon Plouffe, Feb. 20, 2011.
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EXAMPLE
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16*q - 128*q^2 + 704*q^3 - 3072*q^4 + 11488*q^5 - 38400*q^6 + 117632*q^7 - ...
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MATHEMATICA
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a[ n_] := If[ n < 0, 0, SeriesCoefficient[ InverseEllipticNomeQ @ x, {x, 0, n}]]
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ ModularLambda[ Log[q] / (Pi I)], {q, 0, n}]]
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, q] / EllipticTheta[ 3, 0 , q])^4, {q, 0, n}]
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PROG
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(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); 16 * polcoeff( (eta(x + A) * eta(x^4 + A)^2 / eta(x^2 + A)^3)^8, n))}
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CROSSREFS
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Cf. A005798, A014972, A128692.
Sequence in context: A014972 * A128692 A132136 A163399 A067488 A120785
Adjacent sequences: A115974 A115975 A115976 * A115978 A115979 A115980
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KEYWORD
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sign
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AUTHOR
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Michael Somos, Feb 09 2006
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STATUS
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approved
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