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A253243
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Expansion of phi(-x^2) * psi(x^3) * chi(x^3) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.
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4
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1, 0, -2, 2, 0, -4, 1, 0, 0, 2, 0, 0, 3, 0, -4, 2, 0, 0, 2, 0, -2, 0, 0, -4, 2, 0, 0, 2, 0, -4, 1, 0, -4, 4, 0, 0, 0, 0, 0, 2, 0, 0, 3, 0, 0, 2, 0, -4, 2, 0, -4, 0, 0, 0, 4, 0, -2, 2, 0, -4, 2, 0, 0, 0, 0, 0, 0, 0, -8, 2, 0, 0, 1, 0, 0, 4, 0, -4, 2, 0, 0, 2
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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 q^(-1/4) * eta(q^2)^2 * eta(q^6)^4 / (eta(q^3)^2 * eta(q^4) * eta(q^12)) in powers of q.
Euler transform of period 12 sequence [ 0, -2, 2, -1, 0, -4, 0, -1, 2, -2, 0, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 108^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A246650.
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EXAMPLE
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G.f. = 1 - 2*x^2 + 2*x^3 - 4*x^5 + x^6 + 2*x^9 + 3*x^12 - 4*x^14 + 2*x^15 + ...
G.f. = q - 2*q^9 + 2*q^13 - 4*q^21 + q^25 + 2*q^37 + 3*q^49 - 4*q^57 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^2] QPochhammer[ -x^3, x^6] EllipticTheta[ 2, 0, x^(3/2)] / (2 x^(3/8)), {x, 0, n}];
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PROG
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(PARI) {a(n) = if( n<0, 0, n = 4*n + 1; sumdiv(n, d, [ 0, 1, -1, -3, 1, -1, 3, 1, -1] [d%9 + 1]))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^6 + A)^4 / (eta(x^3 + A)^2 * eta(x^4 + A) * eta(x^12 + 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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