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

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

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.

M. D. Hirschhorn, The Power of q, Springer, 2017. See page 172.

Links

Seiichi Manyama, Table of n, a(n) for n = -1..10000

Michael Somos, Introduction to Ramanujan theta functions

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

G.f.: Product_{4|n} (1-q^n)^8. [Hirschhorn] - N. J. A. Sloane, Jan 26 2021

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: A344052 A316201 A029845 * A161969 A000731 A034433

Adjacent sequences: A124969 A124970 A124971 * A124973 A124974 A124975

Keyword

sign

Author

Michael Somos, Nov 14 2006

Status

approved