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A153728
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Expansion of q^(-1/3) * (eta(q)^8 + 8 * eta(q^4)^8) in powers of q^2.
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5
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1, 20, -70, 56, -125, 308, 110, -520, 57, 0, 182, -880, 1190, 884, 0, -1400, -1330, 1820, -646, 0, -1331, 380, 1120, 2576, 0, 1748, -3850, -3400, 2703, -2500, 3458, 0, -1150, -5236, 0, 6032, 6160, -3220, 4466, 0, -7378, -3920, 0, 2200, 0, 812, -4030, 5600, -4913
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OFFSET
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0,2
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COMMENTS
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LINKS
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FORMULA
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a(n) = b(6*n + 1) where b(n) is multiplicative with b(2^e) = b(3^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 * (-1)^(e/2) * p^(3*e/2) if p == 5 (mod 6), b(p^e) = b(p) * b(p^(e-1)) - b(p^(e-2)) * p^3 if p == 1 (mod 6) where b(p) = (x^2 - 3*p)*x, 4*p = x^2 + 3*y^2, |x| < |y| and x == 2 (mod 3).
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 648 (t/i)^4 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A153729.
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EXAMPLE
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G.f. = 1 + 20*x - 70*x^2 + 56*x^3 - 125*x^4 + 308*x^5 + 110*x^6 - 520*x^7 + ...
G.f. = q + 20*q^7 - 70*q^13 + 56*q^19 - 125*q^25 + 308*q^31 + 110*q^37 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ QPochhammer[ x]^8 + 8 x QPochhammer[ x^4]^8, {x, 0, 2 n}]; (* Michael Somos, Jun 10 2015 *)
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PROG
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(PARI) {a(n) = my(A); if( n<0, 0, n *= 2; A = x * O(x^n); polcoeff( eta(x + A)^8 + 8 * x * eta(x^4 + A)^8, n))};
(PARI) {a(n) = my(A, p, e, x, y, a0, a1); if( n<0, 0, n = 6*n + 1; A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p<5, 0, p%6==5, if( e%2, 0, (-p)^(3*e/2)), for(x=1, sqrtint(p\3), if( issquare(p-3*x^2, &y), break)); if( y%3!=1, y=-y); y*=2; y = y^3 - 3*p*y; a0=1; a1=y; for(i=2, e, x = y*a1 - p^3*a0; a0=a1; a1=x); a1)))}; /* Michael Somos, Jun 10 2015 */
(Magma) A := Basis( CuspForms( Gamma0(36), 4), 289); A[1] + 20*A[7] - 70*A[12]; /* Michael Somos, Jun 10 2015 */
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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