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A132985
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Expansion of chi(-q^5) / chi(-q)^5 in powers of q where chi() is a Ramanujan theta function.
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3
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1, 5, 15, 40, 95, 205, 420, 820, 1535, 2785, 4915, 8460, 14260, 23590, 38360, 61440, 97055, 151370, 233355, 355900, 537395, 803960, 1192380, 1754140, 2560980, 3712205, 5344570, 7645600, 10871080, 15368350, 21607220, 30220360, 42056415, 58249680, 80310510
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OFFSET
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0,2
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COMMENTS
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Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
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LINKS
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G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
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Expansion of (eta(q^5) / eta(q^10)) / (eta(q) / eta(q^2))^5 in powers of q.
Euler transform of period 10 sequence [ 5, 0, 5, 0, 4, 0, 5, 0, 5, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 + v + u * v * (2 - 4 * v).
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (u - v)^4 - u * v * (1 + 3 * u - 4 * u^2) * (1 + 3 * v - 4 * v^2).
G.f. is a period 1 Fourier series which satisfies f(-1 / (10 t)) = (1/4) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A132980.
G.f.: Product_{k>0} (1 + x^k)^5 / (1 + x^(5*k)).
a(n) ~ exp(2*Pi*sqrt(2*n/5)) / (2^(11/4) * 5^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2015
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EXAMPLE
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G.f. = 1 + 5*q + 15*q^2 + 40*q^3 + 95*q^4 + 205*q^5 + 420*q^6 + 820*q^7 + ...
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MATHEMATICA
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nmax = 40; CoefficientList[Series[Product[(1 + x^k)^5 / (1 + x^(5*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 07 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ q^5, q^10] / QPochhammer[ q, q^2]^5, {q, 0, n}]; (* Michael Somos, Oct 31 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^5 + A) / eta(x^10 + A) ) / ( eta(x + A) / eta(x^2 + A) )^5, n))};
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CROSSREFS
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Cf. A132980.
Sequence in context: A262295 A034182 A348885 * A022570 A152881 A000333
Adjacent sequences: A132982 A132983 A132984 * A132986 A132987 A132988
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KEYWORD
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nonn
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AUTHOR
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Michael Somos, Sep 07 2007
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EXTENSIONS
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Typo in a(32) corrected by G. C. Greubel, Sep 28 2017
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STATUS
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approved
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