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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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4
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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
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OFFSET
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0,2
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COMMENTS
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Number 44 of the 74 eta-quotients listed in Table I of Martin 1996.
Denoted by g_2(q) in Cynk and Hulek in Remark 3.4 on page 12 as the unique weight 2 newform of level 27.
This is a member of an infinite family of integer weight modular forms. g_1 = A033687, g_2 = A030206, g_3 = A130539, g_4 = A000731. - Michael Somos, Aug 24 2012
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REFERENCES
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M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
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LINKS
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Table of n, a(n) for n=0..89.
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
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Expansion of q^(-1/3) * b(q) * c(q) / 3 in powers of q where b(), c() are cubic AGM theta functions. - Michael Somos, Nov 01 2006
Coefficients of L-series for elliptic curve "27a3": y^2 + y = x^3. - Michael Somos, Aug 13 2006
Euler transform of period 3 sequence [ -2, -2, -4, ...]. - Michael Somos, Dec 06 2004
G.f. is a period 1 Fourier series which satisfies f(-1 / (27 t)) = 27 (t/i)^2 f(t) where q = exp(2 pi i t).
G.f.: Product_{k>0} (1 - x^k)^2 * (1 - x^(3*k))^2.
a(n) = b(3*n + 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
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
a(4*n + 3) = a(16*n + 13) = 0. - Michael Somos, Oct 19 2005
a(4*n + 1) = -2 * a(n). - Michael Somos. Dec 06 2004
a(25*n + 8) = -5 * a(n). Convolution square of A030203. - Michael Somos, Mar 13 2012
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EXAMPLE
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1 - 2*x - x^2 + 5*x^4 + 4*x^5 - 7*x^6 - 5*x^8 + 2*x^9 - 4*x^10 + 11*x^12 + ...
q - 2*q^4 - q^7 + 5*q^13 + 4*q^16 - 7*q^19 - 5*q^25 + 2*q^28 - 4*q^31 + ...
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PROG
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(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
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Cf. A030203.
Sequence in context: A066435 A171960 A182376 * A212768 A133336 A176056
Adjacent sequences: A030203 A030204 A030205 * A030207 A030208 A030209
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KEYWORD
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sign
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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