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A263527
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Expansion of phi(-x^3) * f(-x^6)^3 / f(-x^2) in powers of x where phi(), f() are Ramanujan theta functions.
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4
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1, 0, 1, -2, 2, -2, 0, -4, 2, 0, 1, -4, 4, -2, 2, -4, 5, 0, 2, -2, 6, -4, 2, -4, 6, 0, 0, -6, 4, -2, 4, -8, 7, 0, 2, -10, 4, -6, 0, -4, 6, 0, 1, -6, 8, -6, 4, -8, 4, 0, 4, -8, 10, -4, 2, -8, 8, 0, 2, -6, 12, -4, 4, -8, 8, 0, 5, -8, 6, -4, 0, -8, 14, 0, 2, -10
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OFFSET
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0,4
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COMMENTS
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LINKS
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FORMULA
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Expansion of q^(-2/3) * eta(q^3)^2 * eta(q^6)^2 / eta(q^2) in powers of q.
Euler transform of period 6 sequence [ 0, 1, -2, 1, 0, -3, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = (2048/3)^(1/2) (t/I)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A263501.
a(n) = (-1)^n * A261444(n). a(8*n + 1) = 0.
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EXAMPLE
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G.f. = 1 + x^2 - 2*x^3 + 2*x^4 - 2*x^5 - 4*x^7 + 2*x^8 + x^10 - 4*x^11 + ...
G.f. = q^2 + q^8 - 2*q^11 + 2*q^14 - 2*q^17 - 4*q^23 + 2*q^26 + q^32 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^3] QPochhammer[ x^6]^3 / QPochhammer[ x^2], {x, 0, n}];
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PROG
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(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^2 * eta(x^6 + A)^2 / eta(x^2 + A), n))};
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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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