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A053762
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Number of 3-colored generalized Frobenius partitions of n.
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6
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1, 9, 27, 82, 207, 486, 1055, 2205, 4374, 8427, 15696, 28539, 50630, 88119, 150417, 252727, 418068, 682344, 1099343, 1750968, 2758185, 4301682, 6645150, 10175625, 15451744, 23281686, 34819227, 51712860, 76292784, 111850740, 162997314
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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^(1/8) * (eta(q)^3 + 9 * eta(q^9)^3) / (eta(q)^3 * eta(q^3)) in powers of q. - Michael Somos, Mar 09 2011
Expansion of a(x) / f(-x)^3 in powers of x where a() is a cubic AGM theta function and f() is a Ramanujan theta function. - Michael Somos, Aug 21 2012
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EXAMPLE
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1 + 9*x + 27*x^2 + 82*x^3 + 207*x^4 + 486*x^5 + 1055*x^6 + 2205*x^7 + ...
1/q + 9*q^7 + 27*q^15 + 82*q^23 + 207*q^31 + 486*q^39 + 1055*q^47 + 2205*q^55 + ...
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MATHEMATICA
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nmax = 30; CoefficientList[Series[(Product[(1 - x^k)^3, {k, 1, nmax}] + 9*x*Product[(1 - x^(9*k))^3, {k, 1, nmax}]) / Product[((1 - x^k)^3*(1 - x^(3*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 13 2016 *)
a[n_]:= SeriesCoefficient[q^(1/8)*(eta[q]^3 + 9*eta[q^9]^3)/(eta[q]^3* eta[q^3]), {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Feb 08 2018 *)
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PROG
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(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^3 + 9 * x * eta(x^9 + A)^3) / (eta(x + A)^3 * eta(x^3 + A)), n))} /* Michael Somos, Mar 09 2011 */
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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