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A121589
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Series expansion of (eta(q^9) / eta(q))^3 in powers of q.
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6
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1, 3, 9, 22, 51, 108, 221, 429, 810, 1476, 2631, 4572, 7802, 13056, 21519, 34918, 55935, 88452, 138332, 213990, 327852, 497592, 748833, 1117692, 1655719, 2434938, 3556791, 5161808, 7445631, 10677096, 15226658, 21599469, 30485268, 42817788
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OFFSET
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1,2
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COMMENTS
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LINKS
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FORMULA
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Euler transform of period 9 sequence [3, 3, 3, 3, 3, 3, 3, 3, 0, ...].
G.f.: x * (Product_{k>0} (1 - x^(9*k) / (1 - x^k))^3.
Expansion of c(q^3) / (3 * b(q)) = (c(q) / (3 * b(q^3))^3 in powers of q where b(), c() are cubic AGM functions.
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u - v^2) * (u^2 - v) - 2 * u * v * ( 3 * (u + v) + 13 * u * v ).
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = u^3 - v * (1 + 9 * v + 27 * v^2) * (1 + 9 * u + 27 * u^2).
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u1 * u2 - (1 + 3 * (u1 + u2)) * (u3 + u6 + 9 * u3 * u6).
G.f. is a period 1 Fourier series which satisfies f(-1 / (9 t)) = (1/27) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A131986.
a(n) ~ exp(4*Pi*sqrt(n)/3) / (27 * sqrt(6) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2015
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EXAMPLE
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G.f. = q + 3*q^2 + 9*q^3 + 22*q^4 + 51*q^5 + 108*q^6 + 221*q^7 + 429*q^8 + ...
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MAPLE
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N:= 100: # to get a(1)..a(N)
S:= series(q*Product(1-q^(9*k), k=1..N/9)/Product((1-q^k)^3, k=1..N), q, N+1):
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MATHEMATICA
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nmax = 40; CoefficientList[Series[Product[((1-x^(9*k))/(1-x^k))^3, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 07 2015 *)
QP = QPochhammer; s = (QP[q^9]/QP[q])^3 + O[q]^40; CoefficientList[s, q] (* Jean-François Alcover, Nov 30 2015, adapted from PARI *)
a[ n_] := SeriesCoefficient[ q (QPochhammer[ q^9] / QPochhammer[ q])^3, {q, 0, n}]; (* Michael Somos, Nov 02 2017 *)
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PROG
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(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^9 + A) / eta(x + A))^3, 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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