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A281744
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Expansion of chi(-x^3) * chi(-x^12) / (chi(-x) * chi(-x^4)) in powers of x where chi() is a Ramanujan theta function.
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1
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1, 1, 1, 1, 2, 3, 3, 4, 5, 6, 7, 9, 11, 13, 15, 18, 21, 25, 29, 34, 40, 46, 53, 61, 71, 82, 93, 106, 122, 139, 158, 179, 203, 230, 259, 293, 330, 371, 416, 467, 524, 586, 655, 731, 816, 909, 1011, 1124, 1250, 1387, 1537, 1702, 1885, 2085, 2303, 2543, 2806
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OFFSET
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0,5
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COMMENTS
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The sequence generating function appears on the right side of one of Ramanujan's forty identities.
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LINKS
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FORMULA
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Expansion of G(x) * G(x^24) + x^5 * G(x) * H(x^24) in powers of x where G(), H() are Rogers-Ramanujan functions.
Expansion of q^(5/12) * eta(q^2) * eta(q^3) * eta(q^8) * eta(q^12) / (eta(q) * eta(q^4) * eta(q^6) * eta(q^24)) in powers of q
Euler transform of period 24 sequence [1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, ...].
a(n) ~ 5^(1/4) * exp(sqrt(5*n/2)*Pi/3) / (2^(7/4) * sqrt(3) * n^(3/4)). - Vaclav Kotesovec, Aug 22 2018
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EXAMPLE
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G.f. = 1 + x + x^2 + x^3 + 2*x^4 + 3*x^5 + 3*x^6 + 4*x^7 + 5*x^8 + 6*x^9 + ...
G.f. = q^-5 + q^7 + q^19 + q^31 + 2*q^43 + 3*q^55 + 3*q^67 + 4*q^79 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x] QPochhammer[ -x^4, x^4] QPochhammer[ x^3, x^6] QPochhammer[ x^12, x^24], {x, 0, n}];
nmax = 50; CoefficientList[Series[Product[(1 + x^k) * (1 + x^(4*k)) / ((1 + x^(12*k)) * (1 + x^(3*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 22 2018 *)
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PROG
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(PARI) {a(n) = if( n<0, 0, my(A = x * O(x^n)); polcoeff( eta(x^2 + A) * eta(x^3 + A) * eta(x^8 + A) * eta(x^12 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A) * eta(x^24 + A)), n))};
(PARI) lista(nn) = {q='q+O('q^nn); Vec(eta(q^2)*eta(q^3)*eta(q^8)*eta(q^12)/(eta(q)*eta(q^4)*eta(q^6)*eta(q^24)))} \\ Altug Alkan, Mar 21 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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