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A137828
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Expansion of phi(x) / f(-x^4)^2 in powers of x where phi(), f() are Ramanujan theta functions.
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4
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1, 2, 0, 0, 4, 4, 0, 0, 9, 12, 0, 0, 20, 24, 0, 0, 42, 50, 0, 0, 80, 92, 0, 0, 147, 172, 0, 0, 260, 296, 0, 0, 445, 510, 0, 0, 744, 840, 0, 0, 1215, 1372, 0, 0, 1944, 2176, 0, 0, 3059, 3424, 0, 0, 4740, 5268, 0, 0, 7239, 8040, 0, 0, 10920, 12072, 0, 0, 16286
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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 q^(1/3) * eta(q^2)^5 / (eta(q)^2 * eta(q^4)^4) in powers of q.
Euler transform of period 4 sequence [ 2, -3, 2, 1, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 6^(-1/2) (t/i)^(-1/2) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A051136.
G.f.: Product_{k>0} (1 + x^k) / ( (1 - x^k) * (1 + x^(2*k))^4 ).
a(4*n + 2) = a(4*n + 3) = 0.
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EXAMPLE
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G.f. = 1 + 2*x + 4*x^4 + 4*x^5 + 9*x^8 + 12*x^9 + 20*x^12 + 24*x^13 + 42*x^16 + ...
G.f. = 1/q + 2*q^2 + 4*q^11 + 4*q^14 + 9*q^23 + 12*q^26 + 20*q^35 + 24*q^38 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] / QPochhammer[ x^4]^2, {x, 0, n}]; (* Michael Somos, Oct 04 2015 *)
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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^4 + A)^4 / eta(x + A)^2, 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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