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A030206
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Expansion of q^(-1/3) * eta(q)^2 * eta(q^3)^2 in powers of q.
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0
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1, -2, -1, 0, 5, 4, -7, 0, -5, 2, -4, 0, 11, 0, 8, 0, -6, -10, 0, 0, -1, -8, 5, 0, -7, 14, 17, 0, 0, 0, -5, 0, -19, 10, -13, 0, 2, -4, 0, 0, -11, 8, 20, 0, 7, 0, 23, 0, 0, -22, -19, 0, 14, 0, -25, 0, 12, -16, 5, 0, -7, 0, 0, 0, 23, 12, 11, 0, 0, 20, -13, 0, 4, 0, -28, 0, -22, 0, 0, 0, 17, 2, -35, 0, 0, 16, -11, 0, 0, -10
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Denoted by g_2(q) in Cynk and Hulek in Remark 3.4 on page 12
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REFERENCES
| M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.
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LINKS
| S. R. Finch, Powers of Euler's q-Series, (arXiv:math.NT/0701251).
S. Cynk and K. Hulek, Construction and examples of higher-dimensional modular Calabi-Yau manifolds
W. Stein, Modular Forms Database.
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FORMULA
| Euler transform of period 3 sequence [ -2, -2, -4, ...]. - Michael Somos, Dec 06 2004
a(4n+1)=-2a(n). - Michael Somos Dec 06 2004
G.f.: Product_{k>0} (1-x^k)^2*(1-x^(3k))^2.
a(4n+3)=a(16n+13)=0. - Michael Somos Oct 19 2005
a(n)=b(3n+1) where b(n) is multiplicative and b(3^e) = 0^e, b(p^e) = (1+(-1)^e)/2*(-1)^(e/2)*p^(e/2), if p == 2 (mod 3), b(p^e) = b(p)b(p^(e-1))-p*b(p^(e-2)). - Michael Somos Aug 13 2006
Expansion of q^(-1/3)*b(q)*c(q)/3 in powers of q where b(),c() are cubic AGM analog functions. - Michael Somos Nov 01 2006
Coefficients of L-series for elliptic curve "27a3": y^2 +y= x^3 . - Michael Somos Aug 13 2006
Given g.f. A(x), then B(x)= x*A(x^3) satisfies 0= f(B(x), B(x^2), B(x^4)) where f(u, v, w)= v^3 -u*w*(u +4*w) . - Michael Somos Dec 06 2004
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EXAMPLE
| q - 2*q^4 - q^7 + 5*q^13 + 4*q^16 - 7*q^19 - 5*q^25 + 2*q^28 - ...
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PROG
| (PARI) {a(n)=local(A, p, e, x, y, a0, a1); if(n<0, 0, n=3*n+1; A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==3, 0, if(p%3==2, if(e%2, 0, (-1)^(e/2)*p^(e/2)), for(i=1, sqrtint(4*p\27), if(issquare(4*p-27*i^2, &y), break)); a0=1; a1=y*=(-1)^(y%3); for(i=2, e, x=y*a1-p*a0; a0=a1; a1=x); a1)))))} /* Michael Somos Aug 13 2006 */
(PARI) {a(n)= local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x+A)^2* eta(x^3+A)^2, n))} /* Michael Somos Feb 19 2007 */
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CROSSREFS
| Sequence in context: A113368 A066435 A171960 * A133336 A176056 A130191
Adjacent sequences: A030203 A030204 A030205 * A030207 A030208 A030209
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KEYWORD
| sign
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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