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A113298
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Expansion of q^(1/12) * eta(q^10)^5 / ( eta(q^2) * eta(q^5)^2 * eta(q^20)^2) in powers of q.
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1
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1, 0, 1, 0, 2, 2, 3, 2, 5, 4, 7, 6, 11, 10, 15, 14, 22, 22, 30, 30, 44, 44, 58, 60, 81, 84, 107, 112, 145, 154, 190, 202, 253, 270, 327, 352, 429, 462, 550, 594, 711, 770, 904, 980, 1156, 1256, 1457, 1586, 1845, 2008, 2310, 2516, 2898, 3160, 3604, 3930, 4488, 4894
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OFFSET
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0,5
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COMMENTS
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Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).
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LINKS
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FORMULA
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Euler transform of period 20 sequence [0, 1, 0, 1, 2, 1, 0, 1, 0, -2, 0, 1, 0, 1, 2, 1, 0, 1, 0, 0, ...].
Expansion of phi(q^5) / f(-q^2) in powers of q where phi(), f() are Ramanujan theta functions.
Expansion of G(x) * G(x^4) - x * H(x) * H(x^4) where G() = g.f. A003114 and H() = g.f. A003106 are the Rogers-Ramanujan functions.
G.f.: (1 + 2 * Sum_{k>0} x^(5*k^2)) / (Product_{k>0} (1 - x^(2*k))).
a(n) ~ exp(Pi*sqrt(n/3)) / (2*3^(1/4)*sqrt(10)*n^(3/4)). - Vaclav Kotesovec, Jul 11 2016
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EXAMPLE
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1/q + q^23 + 2*q^47 + 2*q^59 + 3*q^71 + 2*q^83 + 5*q^95 + 4*q^107 + ...
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MATHEMATICA
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nmax = 50; CoefficientList[Series[Product[(1-x^(10*k))^5 / ( (1-x^(2*k)) * (1-x^(5*k))^2 * (1-x^(20*k))^2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 11 2016 *)
eta[q_]:= q^(1/24)*QPochhammer[q]; CoefficientList[Series[q^(1/12)* eta[q^10]^5/(eta[q^2]*eta[q^5]^2*eta[q^20]^2), {q, 0, 50}], q] (* G. C. Greubel, Apr 17 2018 *)
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PROG
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(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^10 + A)^5 / eta(x^2 + A) / eta(x^5 + A)^2 / eta(x^20 + A)^2, n))}
(PARI) q='q+O('q^99); Vec(eta(q^10)^5/(eta(q^2)*eta(q^5)^2*eta(q^20)^2)) \\ Altug Alkan, Apr 18 2018
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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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