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Search: id:A030188
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| A030188 |
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Expansion of q^(-1)eta(q^2)*eta(q^4)*eta(q^6)*eta(q^12) in powers of q^2. |
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2
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| 1, -1, -2, 0, 1, 4, -2, 2, 2, -4, 0, -8, -1, -1, 6, 8, -4, 0, 6, 2, -6, 4, -2, 0, -7, -2, -2, -8, 4, 4, -2, 0, 4, -4, 8, 8, 10, 1, 0, -8, 1, -4, -4, -6, -6, 0, -8, 8, 2, 4, -18, 16, 0, -12, -2, -6, 18, 16, -2, 0, 5, 6, 12, -8, -4, -4, 0, 2, -6, -12, 0, -8, -12, 7, 14, -16, 2, -16, -2, 2, 0, 12, 8, 24, -9, -4, 6, 0, -4, 12, 6, 2, -12, 8, 0, 0
(list; graph; listen)
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OFFSET
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0,3
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REFERENCES
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M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.
J. H. Silverman, A Friendly Introduction to Number Theory, 3rd ed., Pearson Education, Inc, 2006, p. 415. Exer. 47.3.
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LINKS
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W. Stein, Modular Forms Database.
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FORMULA
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Euler transform of period 6 sequence [ -1, -2, -2, -2, -1, -4, ...]. - Michael Somos Apr 2 2005
Given g.f. A(x), then B(x)=x*A(x)^2 satisfies 0=f(B(x), B(x^2), B(x^4)) where f(u, v, w)=u^2vw+4uv^2w+16uvw^2+4u^2w^2-v^4. - Michael Somos Apr 2 2005
a(n)=b(2n+1) where b(n) is multiplicative and b(2^e)=0^e, b(3^e)=(-1)^e, b(p^e) =b(p)*b(p^(e-1)) -p*b(p^(e-2)) otherwise where b(p)= p - number of solutions to y^2=x^3-x^2-4*x+4 modulo p. - Michael Somos Aug 13 2006
G.f.: Product_{k>0} (1-x^k)(1-x^(2k))(1-x^(3k))(1-x^(6k)).
Newform number 1 of degree 1 in Full modular forms space of level 24, weight 2 and trivial character.
G.f. is Fourier series of a weight 2 level 24 modular form. f(-1/ (24 t)) = 24 (t/i)^2 f(t) where q = exp(2 pi i t). - Michael Somos Jun 08 2007
L-series for elliptic curve "24a1": y^2 = x^3 - x^2 - 4*x + 4. - Michael Somos Apr 2 2005
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EXAMPLE
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q - q^3 - 2*q^5 + q^9 + 4*q^11 - 2*q^13 + 2*q^15 + 2*q^17 - 4*q^19 + ...
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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^2+A)*eta(x^3+A)*eta(x^6+A), n))} /* Michael Somos Apr 2 2005 */
(PARI) a(n)=if(n<0, 0, n=2*n+1; ellak(ellinit([0, -1, 0, -4, 4]), n)) /* Michael Somos Apr 2 2005 */
(PARI) {a(n)=local(A, p, e, x, y, a0, a1); if(n<0, 0, n=2*n+1; A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==2, 0, if(p==3, (-1)^e, a0=1; a1=y=-sum(x=0, p-1, kronecker(x^3-x^2-4*x+4, p)); for(i=2, e, x=y*a1-p*a0; a0=a1; a1=x); a1)))))} /* Michael Somos Aug 13 2006 */
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CROSSREFS
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Sequence in context: A124915 A158239 A159819 this_sequence A160648 A124912 A138752
Adjacent sequences: A030185 A030186 A030187 this_sequence A030189 A030190 A030191
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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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