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Search: id:A030208
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| A030208 |
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Expansion of q^(-1/2)(eta(q)*eta(q^3))^3 in powers of q. |
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+0
4
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| 1, -3, 0, 2, 9, 0, -22, 0, 0, 26, -6, 0, 25, -27, 0, -46, 0, 0, 26, 66, 0, -22, 0, 0, -45, 0, 0, 0, -78, 0, 74, 18, 0, 122, 0, 0, -46, -75, 0, -142, 81, 0, 0, 0, 0, -44, 138, 0, 2, 0, 0, 194, 0, 0, -214, -78, 0, 0, -198, 0, 121, 0, 0, 146, 66, 0, 52, 0, 0, -22, 0, 0, 0, 135, 0, -286, 0, 0, -118, 0, 0, -262
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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Expansion of a newform level 12 weight 3 and character [0,1].
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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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Author?, Eta Products and Quotients which are Newforms.
W. Stein, Modular Forms Database.
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FORMULA
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Euler transform of period 3 sequence [ -3,-3,-6,...]. - Michael Somos Feb 13 2006
a(n)=b(2n+1) where b(n) is multiplicative and b(2^e) = 0^e, b(3^e)=(-3)^e, b(p^e) = (1+(-1)^e)/2*p^e if p == 2 (mod 3), b(p^e) = b(p)b(p^(e-1)) -p^2*b(p^(e-2)) otherwise. - Michael Somos Feb 13 2006
G.f.: x*(Product_{k>0} (1-x^k)(1-x^(3k)))^3.
G.f.: Sum_{k>=0} a(k)*x^(2*k+1) = (1/2)* Sum_{u,v} (u*u -3*v*v)* x^(u*u +3*v*v). - Michael Somos Jun 14 2007
a(3*n + 2) = 0. a(3*n + 1) = -3 * a(n). - Michael Somos Nov 16 2008
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 12^(3/2) (t / i)^3 f(t) where q = exp(2 pi i t). - Michael Somos Nov 16 2008
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EXAMPLE
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q -3*q^3 +2*q^7 +9*q^9 -22*q^13 +26*q^19 -6*q^21 +25*q^25 +...
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PROGRAM
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(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( (eta(x+A)*eta(x^3+A))^3, n))} /* Michael Somos Jun 14 2007 */
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CROSSREFS
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Convolution square is A007332. - Michael Somos Nov 16 2008
Sequence in context: A099095 A061980 A059683 this_sequence A126598 A127802 A165951
Adjacent sequences: A030205 A030206 A030207 this_sequence A030209 A030210 A030211
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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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