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A132977
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Expansion of q^(-1/3) * (eta(q^6)^4 / (eta(q) * eta(q^3) * eta(q^4) * eta(q^12)))^2 in powers of q.
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8
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1, 2, 5, 12, 26, 50, 92, 168, 295, 496, 818, 1332, 2126, 3324, 5126, 7824, 11793, 17548, 25857, 37788, 54734, 78578, 111968, 158496, 222842, 311224, 432095, 596676, 819504, 1119624, 1522282, 2060448, 2776514, 3725294, 4978142, 6626988, 8789042
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OFFSET
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0,2
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COMMENTS
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LINKS
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FORMULA
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Expansion of q^(-2/3) * (chi(q) * chi(q^3))^2 * c(q^2) / (3 * b(q^2)) in powers of q where chi() is a Ramanujan theta function and b(), c() are cubic AGM functions.
Euler transform of period 12 sequence [ 2, 2, 4, 4, 2, -4, 2, 4, 4, 2, 2, 0, ...].
Expansion of (chi^3(q^3) / chi(q))^2 * (psi(-q^3) / psi(-q))^4 in powers of q where chi(), psi() are Ramanujan theta functions.
Expansion of q^(-1/3) * (eta(q^6)^4 / (eta(q) * eta(q^3) * eta(q^4) * eta(q^12)))^2 in powers of q.
G.f.: (Product_{k>0} (1-x^(6*k))^4 / ( (1-x^k) * (1-x^(3*k)) * (1-x^(4*k)) * (1-x^(12*k)) ))^2.
a(n) ~ exp(2*Pi*sqrt(n/3)) / (2 * 3^(9/4) * n^(3/4)). - Vaclav Kotesovec, Sep 08 2015
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EXAMPLE
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G.f. = 1 + 2*x + 5*x^2 + 12*x^3 + 26*x^4 + 50*x^5 + 92*x^6 + 168*x^7 + ...
G.f. = q + 2*q^4 + 5*q^7 + 12*q^10 + 26*q^13 + 50*q^16 + 92*q^19 + ...
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MATHEMATICA
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nmax = 40; CoefficientList[Series[Product[((1-x^(6*k))^4 / ( (1-x^k) * (1-x^(3*k)) * (1-x^(4*k)) * (1-x^(12*k)) ))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 08 2015 *)
a[ n_] := SeriesCoefficient[(QPochhammer[ x^6]^4 / (QPochhammer[ x] QPochhammer[ x^3] QPochhammer[ x^4] QPochhammer[ x^12]))^2, {x, 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^6 + A)^4 / (eta(x + A) * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A)))^2, n))};
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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