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Search: id:A030203
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| A030203 |
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Expansion of q^(-1)*eta(q^6)*eta(q^18) in powers of q^6. |
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+0
1
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| 1, -1, -1, -1, 1, 2, -1, 2, 0, 0, -1, -1, -1, -1, 0, 1, -1, -1, 2, 0, 1, 2, 1, -1, 0, -1, 2, -1, 0, -1, -1, 0, -1, -1, 0, -1, -2, 2, 2, 0, -1, 1, 0, 1, 0, -1, 2, 2, 1, 0, -2, 2, -1, 0, -1, -1, -1, 1, -1, 0, 0, -1, -1, -1, 0, 0, 2, -2, -1, 0, -1, 1, 2, 2, 0, 0, 2, -1, 1, -1, 1, -1, 0, 2, 0, 1, 0, -1, 1, 0, -1, -1, 1, -2, 0, -1, -1, 0, -2, 0, 2, -1, -1, -1, 1
(list; graph; listen)
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OFFSET
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0,6
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REFERENCES
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M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.
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LINKS
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S. R. Finch, Powers of Euler's q-Series, (arXiv:math.NT/0701251).
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FORMULA
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Euler transform of period 3 sequence [ -1, -1, -2, ...]. - Michael Somos Jul 27 2006
Expansion of f(-q)f(-q^3) where f(-q)=f(-q,-q^2) is a Ramanujan theta function. - Michael Somos Jul 27 2006
G.f.: Product_{k>0} (1-x^k)(1-x^(3k)).
Given g.f. A(x), then B(x)=(xA(x^6))^2 satisfies 0=f(B(x), B(x^2), B(x^4)) where f(u,v,w)=v^3 -u^2*w -4*u*w^2 . - Michael Somos Jul 27 2006
a(n)=b(6n+1) where b(n) is multiplicative and b(2^e)=b(3^e)=0^e, b(p^e)=(1+(-1)^e)/2 if p == 5 (mod 6), b(p^e) = e+1 if p=x^2+27y^2, b(p^e) = [1,-1,0] depending on e (mod 3) otherwise.
Expansion of q^(-1/6)*sqrt(b(q)*c(q)/3) in powers of q where b(),c() are cubic AGM analog functions. - Michael Somos Nov 01 2006
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EXAMPLE
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q - q^7 - q^13 - q^19 + q^25 + 2*q^31 - q^37 + 2*q^43 - q^61 - ...
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PROGRAM
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(PARI) {a(n)= local(A, p, e); if(n<0, 0, n=6*n+1; A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p<5, 0, if(p%6==5, (1+(-1)^e)/2, if((p-1)/znorder(Mod(2, p))%3, kronecker(e+1, 3), e+1))))))} /* Michael Somos Jul 27 2006 */
(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x+A)*eta(x^3+A), n))} /* Michael Somos Jul 27 2006 */
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CROSSREFS
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Sequence in context: A035186 A035194 A161491 this_sequence A101664 A091952 A108803
Adjacent sequences: A030200 A030201 A030202 this_sequence A030204 A030205 A030206
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KEYWORD
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sign
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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