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.: q^(-1)*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.
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
KEYWORD
sign
AUTHOR
Michael Somos, Nov 14 2006
STATUS
approved