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A030188
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Expansion of (1/q) * eta(q^2) * eta(q^4) * eta(q^6) * eta(q^12) in powers of q^2.
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1
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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; refs; listen; history; internal format)
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OFFSET
| 0,3
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REFERENCES
| 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
| W. Stein, Modular Forms Database.
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FORMULA
| 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^2 * v * w + 4 * u * v^2 * w + 16 * u * v * w^2 + 4 * u^2 * w^2 - v^4. - Michael Somos Apr 2 2005
a(n) = b(2*n + 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+1 - number of solutions to y^2 = x^3 - x^2 - 4*x + 4 modulo p including the point at infinity. - Michael Somos Mar 04 2011
G.f.: Product_{k>0} (1 - x^k) * (1 - x^(2*k)) * (1 - x^(3*k)) * (1 - x^(6*k)).
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
Coefficients of 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
| 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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PROG
| (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], 1), 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
| Sequence in context: A124915 A158239 A159819 * A176703 A160648 A124912
Adjacent sequences: A030185 A030186 A030187 * A030189 A030190 A030191
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KEYWORD
| sign
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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