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A124972 Expansion of Fricke's 32*tau_4(z) in powers of q = exp(2*Pi*i*z). 4

%I

%S 1,-8,20,0,-62,0,216,0,-641,0,1636,0,-3778,0,8248,0,-17277,0,34664,0,

%T -66878,0,125312,0,-229252,0,409676,0,-716420,0,1230328,0,-2079227,0,

%U 3460416,0,-5677816,0,9198424,0,-14729608,0,23328520,0,-36567242,0,56774712,0

%N Expansion of Fricke's 32*tau_4(z) in powers of q = exp(2*Pi*i*z).

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%C 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.

%C Number 1 of the 15 generalized eta-quotients listed in Table I of Yang 2004. - _Michael Somos_, Jul 21 2014

%C A generator (Hauptmodul) of the function field associated with congruence subgroup Gamma_0(4). [Yang 2004] - _Michael Somos_, Jul 21 2014

%D R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Teubner, 1922, Vol. 2, see pp. 373-375.

%H Seiichi Manyama, <a href="/A124972/b124972.txt">Table of n, a(n) for n = -1..10000</a>

%H M. Somos, <a href="http://somos.crg4.com/multiq.html">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%H Y. Yang, <a href="http://dx.doi.org/10.1112/S0024609304003510">Transformation formulas for generalized Dedekind eta functions</a>, Bull. London Math. Soc. 36 (2004), no. 5, 671-682. See p. 679, Table 1.

%F Expansion of (eta(q) / eta(q^4))^8 in powers of q.

%F Expansion of (chi(-q) * chi(-q^2))^8 / q in powers of q where chi() is a Ramanujan theta function.

%F Expansion of -16 + 16 / lambda(z) in powers of nome q = exp(pi*i*z).

%F Euler transform of period 4 sequence [ -8, -8, -8, 0, ...].

%F G.f. A(q) satisfies 0 = f(A(q), A(q^2)) where f(u, v) = u * (16 + u) * (16 + v) - v^2.

%F 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.

%F Elliptic j(z) = 64 * (x^2 + 8*x + 4)^3 / (x^4 * (2*x + 1)) where x = tau_4(z).

%F tau_4(-1 / (4*z)) = 1/(4*tau_4(z)).

%F G.f.: 1/x * (Product_{k>0} (1 - x^k) / (1 - x^(4*k)))^8.

%F a(n) = A029845(n) unless n=0. a(2*n-1) = A007248(n). a(2*n) = 0 unless n=0.

%F Convolution inverse is A092877.

%F 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

%e G.f. = 1/q - 8 + 20*q - 62*q^3 + 216*q^5 - 641*q^7 + 1636*q^9 - 3778*q^11 + ...

%t a[ n_] := SeriesCoefficient[ 16 (-1 + 1 / ModularLambda[ Log[q] / (Pi I)]), {q, 0, n}]; (* _Michael Somos_, Jun 13 2012 *)

%t 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 *)

%t a[ n_] := SeriesCoefficient[ 1/x (QPochhammer[ x] / QPochhammer[ x^4])^8, {x, 0, n}]; (* _Michael Somos_, Dec 15 2016 *)

%o (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))};

%Y Cf. A007248, A029845, A092877.

%K sign

%O -1,2

%A _Michael Somos_, Nov 14 2006

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Last modified January 16 21:37 EST 2019. Contains 319206 sequences. (Running on oeis4.)