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A258065
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Expansion of (phi(-x^3) * f(-x^2))^2 in powers of x where phi(), f() are Ramanujan theta functions.
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1
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1, 0, -2, -4, -1, 8, 6, 4, -7, -8, -2, -4, 10, -8, -4, 0, 2, 16, -2, 16, 5, -8, 0, -12, -12, -16, -2, 12, -9, 0, 6, 8, 2, 16, 12, -20, 0, -8, 22, 0, 18, 8, -32, 0, 4, 8, -26, -28, -13, -8, 0, 12, -6, 24, 2, 20, 18, 0, 30, -16, -3, -8, -10, 20, 0, -16, 14, -16
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OFFSET
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0,3
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COMMENTS
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LINKS
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FORMULA
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Expansion of (f(x, x^2) * f(-x))^2 in powers of x where f() is the Ramanujan general theta function.
Expansion of q^(-1/6) * (eta(q^2) * eta(q^3)^2 / eta(q^6))^2 in powers of q.
Euler transform of period 6 sequence [ 0, -2, -4, -2, 0, -4, ...].
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EXAMPLE
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G.f. = 1 - 2*x^2 - 4*x^3 - x^4 + 8*x^5 + 6*x^6 + 4*x^7 - 7*x^8 - 8*x^9 + ...
G.f. = q - 2*q^13 - 4*q^19 - q^25 + 8*q^31 + 6*q^37 + 4*q^43 - 7*q^49 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, x^3] QPochhammer[ x^2])^2, {x, 0, n}];
a[ n_] := SeriesCoefficient[ (QPochhammer[ x^2] QPochhammer[ x^3]^2 / QPochhammer[ x^6])^2, {x, 0, n}];
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PROG
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(PARI) {a(n) = if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^3 + A)^2 / eta(x^6 + A))^2, n))};
(Magma) A := Basis( ModularForms( Gamma0(72), 2), 409); A[2] - 2*A[14];
(Magma) A := Basis( CuspForms( Gamma0(72), 2), 409); A[1];
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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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