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A124972
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Expansion of Fricke 32*tau_4(z) in powers of q = exp(2*pi*i*z).
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2
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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; internal format)
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OFFSET
| -1,2
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COMMENTS
| Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A10054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (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.
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REFERENCES
| R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Teubner, 1922, Vol. 2, see pp. 373-375.
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LINKS
| M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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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(x) satisfies 0=f(A(x), A(x^2)) where f(u, v) = u*(16+u)*(16+v) -v^2.
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^(4k)))^8.
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EXAMPLE
| 1/q - 8 + 20*q - 62*q^3 + 216*q^5 - 641*q^7 + 1636*q^9 - 3778*q^11 + ...
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PROG
| (PARI) {a(n)=local(A); if(n<-1, 0, n++; A=x*O(x^n); polcoeff( (eta(x+A)/eta(x^4+A))^8, n))}
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CROSSREFS
| A007248(n)=a(2n-1). A029845(n)=a(n) except n=0.
Sequence in context: A177124 A153704 A029845 * A000731 A161969 A034433
Adjacent sequences: A124969 A124970 A124971 * A124973 A124974 A124975
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KEYWORD
| sign
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AUTHOR
| Michael Somos, Nov 14 2006
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