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A283000
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Expansion of chi(-x)^2 * chi(x^3)^2 * chi(-x^4) / chi(x^6) in powers of x where chi() is a Ramanujan theta function.
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1
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1, -2, 1, 0, -1, 0, 0, 2, 1, 0, 0, 0, -1, -4, -1, 0, 2, 0, 1, 6, -2, 0, -1, 0, 2, -8, 1, 0, -3, 0, -1, 12, 4, 0, 2, 0, -5, -18, -2, 0, 5, 0, 2, 24, -6, 0, -3, 0, 8, -32, 4, 0, -9, 0, -4, 44, 10, 0, 4, 0, -12, -58, -6, 0, 15, 0, 7, 76, -17, 0, -7, 0, 19, -100
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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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Expansion of phi(x^3) * chi(-x)^2 / f(x^4, x^8) in powers of x where phi(), chi(), f() are Ramanujan theta functions.
Expansion of f(-x, -x^5)^2 / (f(x^4, x^8) * f(x^6, x^18)) in powers of x where f(, ) is Ramanujan's general theta functions.
Expansion of q^(1/4) * eta(q)^2 * eta(q^4) * eta(q^6)^5 * eta(q^24) / (eta(q^2)^2 * eta(q^3)^2 *eta(q^8) * eta(q^12)^4) in powers of q.
Euler transform of period 24 sequence [-2, 0, 0, -1, -2, -3, -2, 0, 0, 0, -2, 0, -2, 0, 0, 0, -2, -3, -2, -1, 0, 0, -2, 0, ...].
G.f.: Product_{k>0} (1 - x^k + x^(2*k))^2 * (1 - x^(4*k) + x^(8*k)) / (1 + x^(6*k))^3.
a(n) = A134178(2*n + 1). a(6*n + 3) = a(6*n + 5) = 0.
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EXAMPLE
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G.f. = 1 - 2*x + x^2 - x^4 + 2*x^7 + x^8 - x^12 - 4*x^13 - x^14 + 2*x^16 + ...
G.f. = q^-1 - 2*q^3 + q^7 - q^15 + 2*q^27 + q^31 - q^47 - 4*q^51 - q^55 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2]^2 QPochhammer[ -x^3, x^6]^2 QPochhammer[ x^4, x^8] QPochhammer[ x^6, -x^6], {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 + A)^2 * eta(x^4 + A) * eta(x^6 + A)^5 * eta(x^24 + A) / (eta(x^2 + A)^2 * eta(x^3 + A)^2 * eta(x^8 + A) * eta(x^12 + A)^4), n))};
(PARI) lista(nn) = {q='q+O('q^nn); Vec(eta(q)^2*eta(q^4)*eta(q^6)^5*eta(q^24)/(eta(q^2)^2*eta(q^3)^2 *eta(q^8)*eta(q^12)^4))} \\ Altug Alkan, Mar 21 2018
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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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