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A229180
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Expansion of (chi(-x) * chi(-x^3))^-3 in powers of x where chi() is a Ramanujan theta function.
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3
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1, 3, 6, 16, 33, 60, 118, 210, 354, 612, 1008, 1608, 2583, 4035, 6174, 9448, 14196, 21024, 31054, 45282, 65322, 93884, 133638, 188640, 265225, 370086, 512934, 708136, 971628, 1325724, 1802134, 2437200, 3280452, 4400132, 5876184, 7815288, 10360890, 13683525
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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).
In Verrill (1999) section 2.6, denoted by g as a function of q.
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REFERENCES
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H. Verrill, Some Congruences related to modular forms, Max Planck Institute, 1999.
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LINKS
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G. C. Greubel, Table of n, a(n) for n = 0..2500
Michael Somos, Introduction to Ramanujan theta functions
H. Verrill, Some Congruences related to modular forms
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
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Expansion of q^(-1/2) * (eta(q^2) * eta(q^6) / (eta(q) * eta(q^3)))^3 in powers of q.
Euler transform of period 6 sequence [3, 0, 6, 0, 3, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = (1/8) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A058492.
G.f.: t / (1 - 10*t^2 + 9*t^4)^(1/2) where t = the g.f. of A217786.
G.f.: 1 / (Product_{k>0} (1 - x^(2*k - 1)) * (1 - x^(6*k - 3)))^3.
Convolution inverse of A058492.
a(n) ~ exp(2*Pi*sqrt(n/3)) / (16 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2015
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EXAMPLE
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G.f. = 1 + 3*x + 6*x^2 + 16*x^3 + 33*x^4 + 60*x^5 + 118*x^6 + 210*x^7 + ...
G.f. = q + 3*q^3 + 6*q^5 + 16*q^7 + 33*q^9 + 60*q^11 + 118*q^13 + 210*q^15 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ 1 / (QPochhammer[ x, x^2] QPochhammer[x^3, x^6])^3, {x, 0, n}];
nmax = 40; CoefficientList[Series[Product[1/((1 - x^(2*k - 1)) * (1 - x^(6*k - 3)))^3, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 07 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) * eta(x^6 + A) / (eta(x + A) * eta(x^3 + A)))^3, n))};
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CROSSREFS
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Cf. A058492, A084471, A217786.
Sequence in context: A291611 A308401 A196261 * A219810 A122742 A052370
Adjacent sequences: A229177 A229178 A229179 * A229181 A229182 A229183
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KEYWORD
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nonn
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AUTHOR
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Michael Somos, Sep 30 2013
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STATUS
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approved
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