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A260414
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Expansion of psi(x^3) * psi(x^6) / f(-x^4) in powers of x where phi(), f() are Ramanujan theta functions.
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1
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1, 0, 0, 1, 1, 0, 1, 1, 2, 2, 1, 2, 3, 2, 2, 4, 5, 4, 5, 6, 7, 7, 7, 9, 12, 11, 11, 15, 16, 16, 18, 21, 24, 25, 26, 31, 36, 35, 38, 45, 50, 51, 55, 63, 69, 73, 77, 87, 98, 101, 107, 122, 132, 138, 149, 164, 180, 190, 201, 222, 243, 254, 271, 300, 324, 340, 364
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OFFSET
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0,9
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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
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016.
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 q^(-23/24) * eta(q^6) * eta(q^12)^2 / (eta(q^3) * eta(q^4)) in powers of q.
Euler transform of period 12 sequence [ 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, -1, ...].
a(n) ~ exp(Pi*sqrt(n/6)) / (12*sqrt(n)). - Vaclav Kotesovec, Jul 11 2016
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EXAMPLE
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G.f. = 1 + x^3 + x^4 + x^6 + x^7 + 2*x^8 + 2*x^9 + x^10 + 2*x^11 + ...
G.f. = q^23 + q^95 + q^119 + q^167 + q^191 + 2*q^215 + 2*q^239 + q^263 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x^(3/2)] EllipticTheta[ 2, 0, x^3] / (4 x^(9/8) QPochhammer[ x^4]), {x, 0, n}];
nmax = 100; CoefficientList[Series[Product[(1+x^(3*k)) * (1-x^(12*k))^2 / (1-x^(4*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 11 2016 *)
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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^6 + A) * eta(x^12 + A)^2 / (eta(x^3 + A) * eta(x^4 + A)), n))};
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CROSSREFS
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Sequence in context: A294186 A294185 A035462 * A160735 A216338 A227725
Adjacent sequences: A260411 A260412 A260413 * A260415 A260416 A260417
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KEYWORD
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nonn
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AUTHOR
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Michael Somos, Jul 24 2015
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STATUS
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approved
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