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 A124972 Expansion of Fricke's 32*tau_4(z) in powers of q = exp(2*Pi*i*z). 4
 1, -8, 20, 0, -62, 0, 216, 0, -641, 0, 1636, 0, -3778, 0, 8248, 0, -17277, 0, 34664, 0, -66878, 0, 125312, 0, -229252, 0, 409676, 0, -716420, 0, 1230328, 0, -2079227, 0, 3460416, 0, -5677816, 0, 9198424, 0, -14729608, 0, 23328520, 0, -36567242, 0, 56774712, 0 (list; graph; refs; listen; history; text; internal format)
 OFFSET -1,2 COMMENTS Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). Fricke denotes tau_4(omega) the unique period one one-to-one function on polygon T_4 (all omega with real part absolute value less than one-half and above circles with radius one-quarter centered at one-quarter and minus one-quarter) whose value at zero is zero, at one-half is minus one-half and at infinity is infinity. See page 373 equation (11) and paragraph before it. Number 1 of the 15 generalized eta-quotients listed in Table I of Yang 2004. - Michael Somos, Jul 21 2014 A generator (Hauptmodul) of the function field associated with congruence subgroup Gamma_0(4). [Yang 2004] - Michael Somos, Jul 21 2014 REFERENCES R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Teubner, 1922, Vol. 2, see pp. 373-375. LINKS Seiichi Manyama, Table of n, a(n) for n = -1..10000 Eric Weisstein's World of Mathematics, Ramanujan Theta Functions Y. Yang, Transformation formulas for generalized Dedekind eta functions, Bull. London Math. Soc. 36 (2004), no. 5, 671-682. See p. 679, Table 1. FORMULA Expansion of (eta(q) / eta(q^4))^8 in powers of q. Expansion of (chi(-q) * chi(-q^2))^8 / q in powers of q where chi() is a Ramanujan theta function. Expansion of -16 + 16 / lambda(z) in powers of nome q = exp(pi*i*z). Euler transform of period 4 sequence [ -8, -8, -8, 0, ...]. G.f. A(q) satisfies 0 = f(A(q), A(q^2)) where f(u, v) = u * (16 + u) * (16 + v) - v^2. G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 256 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A092877. Elliptic j(z) = 64 * (x^2 + 8*x + 4)^3 / (x^4 * (2*x + 1)) where x = tau_4(z). tau_4(-1 / (4*z)) = 1/(4*tau_4(z)). G.f.: 1/x * (Product_{k>0} (1 - x^k) / (1 - x^(4*k)))^8. a(n) = A029845(n) unless n=0. a(2*n-1) = A007248(n). a(2*n) = 0 unless n=0. Convolution inverse is A092877. a(-1) = 1, a(n) = -(8/(n+1))*Sum_{k=1..n+1} A046897(k)*a(n-k) for n > -1. - Seiichi Manyama, Mar 29 2017 EXAMPLE G.f. = 1/q - 8 + 20*q - 62*q^3 + 216*q^5 - 641*q^7 + 1636*q^9 - 3778*q^11 + ... MATHEMATICA a[ n_] := SeriesCoefficient[ 16 (-1 + 1 / ModularLambda[ Log[q] / (Pi I)]), {q, 0, n}]; (* Michael Somos, Jun 13 2012 *) a[ n_] := With[ {m = n + 1}, SeriesCoefficient[ ( Product[ 1 - q^k, {k, m}] / Product[ 1 - q^k, {k, 4, m, 4}])^8, {q, 0, m}]]; (* Michael Somos, Jun 13 2012 *) a[ n_] := SeriesCoefficient[ 1/x (QPochhammer[ x] / QPochhammer[ x^4])^8, {x, 0, n}]; (* Michael Somos, Dec 15 2016 *) PROG (PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^4 + A))^8, n))}; CROSSREFS Cf. A007248, A029845, A092877. Sequence in context: A153704 A316201 A029845 * A161969 A000731 A034433 Adjacent sequences:  A124969 A124970 A124971 * A124973 A124974 A124975 KEYWORD sign AUTHOR Michael Somos, Nov 14 2006 STATUS approved

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Last modified October 17 06:08 EDT 2019. Contains 328106 sequences. (Running on oeis4.)