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A029845
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Expansion of 16/lambda(z) in powers of nome q = exp(pi*i*z).
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5
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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).
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REFERENCES
| J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339.
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LINKS
| M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Eric Weisstein's World of Mathematics, Elliptic Lambda Function
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FORMULA
| Expansion of (eta(q^2)^3/(eta(q)*eta(q^4)^2))^8 in powers of q. - Michael Somos Nov 14 2006
Expansion of (chi(q)*chi(-q^2))^8/q in powers of q where chi() is a Ramanujan theta function.
Euler transform of period 4 sequence [ 8, -16, 8, 0, ...]. - Michael Somos Nov 14 2006
G.f. A(x) satisfies 0=f(A(x), A(x^2)) where f(u, v) = 256 -v*(32-16*u+u^2) +v^2 . - Michael Somos Nov 14 2006
G.f.: 1/x*(Product_{k>0} (1+q^(2k-1))/(1+q^(2k)))^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( 16*x+(eta(x+A)/eta(x^4+A))^8, n))} /* Michael Somos Nov 14 2006 */
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CROSSREFS
| Cf. A007248.
A007248(n)=a(2n-1). A124972(n)=a(n) except n=0.
Sequence in context: A177124 A153704 * A124972 A000731 A161969 A034433
Adjacent sequences: A029842 A029843 A029844 * A029846 A029847 A029848
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KEYWORD
| sign,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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