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A164269
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Expansion of q * f(q^9)^3 * phi(q) / (f(q^3) * phi(q^3)^3) in powers of q where f(), phi() are Ramanujan theta functions.
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4
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1, 2, 0, -7, -12, 0, 32, 50, 0, -114, -168, 0, 350, 496, 0, -967, -1332, 0, 2468, 3324, 0, -5916, -7824, 0, 13471, 17548, 0, -29384, -37788, 0, 61784, 78578, 0, -125838, -158496, 0, 249230, 311224, 0, -481506, -596676, 0, 909788, 1119624, 0, -1684824, -2060448, 0
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OFFSET
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1,2
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COMMENTS
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LINKS
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FORMULA
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Euler transform of period 36 sequence [2, -3, -5, -1, 2, 8, 2, -1, -2, -3, 2, 3, 2, -3, -5, -1, 2, 2, 2, -1, -5, -3, 2, 3, 2, -3, -2, -1, 2, 8, 2, -1, -5, -3, 2, 0, ...].
Expansion of x * phi(x) * chi(x^3) * f(x^9)^3 / phi(x^3)^4 = x * phi(x) * f(x^9)^3 / (chi(x^3)^3 * f(x^3)^4) in powers of x where phi(), chi(), f() are Ramanujan theta functions. - Michael Somos, Sep 20 2017
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EXAMPLE
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G.f. = q + 2*q^2 - 7*q^4 - 12*q^5 + 32*q^7 + 50*q^8 - 114*q^10 - 168*q^11 + ...
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MATHEMATICA
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f[x_, y_] := QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; A164269[n_] := SeriesCoefficient[q*(f[q^9, -q^18]^3*f[q, q])/(( f[q^3, -q^6])*f[q^3, q^3]^3), {q, 0, n}]; Rest[Table[A164269[n], {n, 0, 50}]] (* G. C. Greubel, Sep 16 2017 *)
a[ n_] := SeriesCoefficient[ x QPochhammer[ -x^9]^3 EllipticTheta[ 3, 0, x] / (QPochhammer[ -x^3] EllipticTheta[ 3, 0, x^3]^3), {x, 0, n}]; (* Michael Somos, Sep 17 2017 *)
a[ n_] := SeriesCoefficient[ x QPochhammer[ -x^3, x^6] QPochhammer[ x^9]^3 EllipticTheta[ 3, 0, x] / EllipticTheta[ 3, 0, x^3]^4, {x, 0, n}]; /(* Michael Somos, Sep 20 2017 *)
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PROG
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(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^3 + A)^7 * eta(x^12 + A)^7 * eta(x^18 + A)^9 / (eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^6 + A)^18 * eta(x^9 + A)^3 * eta(x^36 + A)^3), 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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