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A260109
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Expansion of f(x^3) * f(-x^3)^2 * psi(x)^2 / psi(-x) in powers of x where psi(), f() are Ramanujan theta functions.
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3
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1, 3, 4, 6, 9, 12, 14, 12, 16, 18, 18, 24, 21, 27, 28, 30, 36, 24, 38, 42, 40, 42, 36, 48, 43, 48, 52, 48, 54, 60, 62, 54, 56, 66, 72, 72, 74, 63, 72, 78, 81, 84, 64, 84, 88, 84, 90, 72, 98, 108, 100, 102, 72, 108, 110, 114, 112, 96, 126, 96, 133, 120, 104
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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 psi(-x^3) * phi(-x^6)^2 * psi(x)^2 / psi(-x) in powers of x where phi(), psi() are Ramanujan theta functions.
Expansion of phi(x) * f(x, x^5) * f(x^2, x^4)^2 in powers of x. - Michael Somos, Jul 18 2015
Expansion of q^(-1/2) * eta(q^2)^5 * eta(q^3) * eta(q^6)^3 / (eta(q)^3 * eta(q^4) * eta(q^12)) in powers of q.
Euler transform of period 12 sequence [ 3, -2, 2, -1, 3, -6, 3, -1, 2, -2, 3, -4, ...].
a(n) = A124815(2*n + 1). a(3*n + 1) = 3 * a(n).
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EXAMPLE
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G.f. = 1 + 3*x + 4*x^2 + 6*x^3 + 9*x^4 + 12*x^5 + 14*x^6 + 12*x^7 + ...
G.f. = q + 3*q^3 + 4*q^5 + 6*q^7 + 9*q^9 + 12*q^11 + 14*q^13 + 12*q^15 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ 1/4 x^(-1/2) EllipticTheta[ 2, 0, x^(1/2)]^2 EllipticTheta[ 2, Pi/4, x^(3/2)] EllipticTheta[ 4, 0, x^6]^2 / EllipticTheta[ 2, Pi/4, x^(1/2)], {x, 0, n}];
a[ n_] := SeriesCoefficient[ 2^(-3/2) x^(-1/8) QPochhammer[ -x^3] QPochhammer[ x^3]^2 EllipticTheta[ 2, 0, x^(1/2)]^2 / EllipticTheta[ 2, Pi/4, x^(1/2)], {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^2 + A)^5 * eta(x^3 + A) * eta(x^6 + A)^3 / (eta(x + A)^3 * eta(x^4 + A) * eta(x^12 + A)), n))};
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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