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A115977 Expansion of elliptic modular function lambda in powers of the nome q.

%I A115977 #61 Feb 16 2025 08:33:00
%S A115977 16,-128,704,-3072,11488,-38400,117632,-335872,904784,-2320128,
%T A115977 5702208,-13504512,30952544,-68901888,149403264,-316342272,655445792,
%U A115977 -1331327616,2655115712,-5206288384,10049485312,-19115905536,35867019904,-66437873664
%N A115977 Expansion of elliptic modular function lambda in powers of the nome q.
%C A115977 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D A115977 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 591.
%D A115977 J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 121.
%D A115977 A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, p. 23, eq. (37).
%H A115977 Seiichi Manyama, <a href="/A115977/b115977.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from G. C. Greubel)
%H A115977 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H A115977 A. Dieckmann, <a href="http://www-elsa.physik.uni-bonn.de/~dieckman/InfProd/InfProd.html">Collection of Infinite Products and Series</a>
%H A115977 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A115977 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H A115977 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EllipticLambdaFunction.html">Elliptic Lambda Function</a>
%H A115977 Wolfram Research <a href="http://functions.wolfram.com/EllipticFunctions/EllipticTheta1/18/01/01/">Basic Algebraic Identities</a> Relations involving squares, 1st formula
%F A115977 Expansion of Jacobi elliptic parameter m = k^2 = (theta_2(q) / theta_3(q))^4 in powers of the nome q.
%F A115977 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.
%F A115977 Expansion of 16 * (eta(q) * eta(q^4)^2 / eta(q^2)^3)^8 in powers of q.
%F A115977 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).
%F A115977 lambda( -1 / tau ) = 1 - lambda( tau ) (see A128692).
%F A115977 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 the g.f. for A128692.
%F A115977 G.f.: 16 * q * (Product_{k>0} (1 + q^(2*k)) / (1 + q^(2*k - 1)))^8.
%F A115977 a(n) = 16 * A005798(n). a(n) = -(-1)^n * A014972(n) unless n=0.
%F A115977 a(n) = -(-1)^n * A132136(n). - _Michael Somos_, Jun 03 2015
%F A115977 Empirical: Sum_{n>=1}(exp(-2*Pi)^n*a(n)) = 17 - 12*sqrt(2). - _Simon Plouffe_, Feb 20 2011
%F A115977 a(n) ~ -(-1)^n * exp(2*Pi*sqrt(n)) / (32 * n^(3/4)). - _Vaclav Kotesovec_, Apr 06 2018
%F A115977 The g.f. A(q) = 16*q - 128*q^2 + 704*q^3 - 3072*q^4 + ... satisfies A(q) + A(-q) = A(q)*A(-q). - _Peter Bala_, Sep 26 2023
%e A115977 G.f. = 16*q - 128*q^2 + 704*q^3 - 3072*q^4 + 11488*q^5 - 38400*q^6 + 117632*q^7 - ...
%t A115977 a[ n_] := SeriesCoefficient[ InverseEllipticNomeQ @ x, {x, 0, n}];
%t A115977 a[ n_] := If[ n < 0, 0, SeriesCoefficient[ ModularLambda[ Log[q] / (Pi I)], {q, 0, n}]];
%t A115977 a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, q] / EllipticTheta[ 3, 0, q])^4, {q, 0, n}];
%t A115977 a[ n_] := SeriesCoefficient[ 1/16 (EllipticTheta[ 2, 0, q] / EllipticTheta[ 3, 0, q^2])^8, {q, 0, n}]; (* _Michael Somos_, May 26 2016 *)
%o A115977 (PARI) {a(n) = my(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))};
%Y A115977 Cf. A005798, A014972, A128692, A132136.
%K A115977 sign
%O A115977 1,1
%A A115977 _Michael Somos_, Feb 09 2006