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A123633
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Expansion of (c(q^2)/c(q))^3 in powers of q where c() is a cubic AGM theta function.
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6
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1, -3, 3, 5, -18, 15, 24, -75, 57, 86, -252, 183, 262, -744, 522, 725, -1998, 1365, 1852, -4986, 3336, 4436, -11736, 7719, 10103, -26322, 17067, 22040, -56682, 36306, 46336, -117867, 74700, 94378, -237744, 149277, 186926, -466836, 290706, 361126, -895014, 553224
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OFFSET
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1,2
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COMMENTS
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In the arXiv:2305.13951 paper on page 21 is this: "The q-expansion of y coincides with the sequence A123633 in the OEIS". - Michael Somos, May 26 2023
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LINKS
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FORMULA
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Expansion of q / (chi(-q^3)^3 / chi(-q))^3 in powers of q where chi() is a Ramanujan theta function.
Euler transform of period 6 sequence [ -3, 0, 6, 0, -3, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v)= u^2 - v - u*v * (6 + 8*v).
G.f.: x * (Product_{k>0} (1 - x^(2*k - 1)) / (1 - x^(6*k - 3))^3 )^3.
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = (1 / 8) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A128642.
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EXAMPLE
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G.f. = q - 3*q^2 + 3*q^3 + 5*q^4 - 18*q^5 + 15*q^6 + 24*q^7 - 75*q^8 + 57*q^9 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ q / (QPochhammer[ q^3, q^6]^3 / QPochhammer[ q, q^2])^3, {q, 0, n}]; (* Michael Somos, Feb 19 2015 *)
a[ n_] := SeriesCoefficient[ q (Product[ 1 - q^k, {k, 1, n, 2}] / Product[ 1 - q^k, {k, 3, n, 6}]^3)^3, {q, 0, n}]; (* Michael Somos, Feb 19 2015 *)
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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 + A) / eta(x^2 + A))^3 * (eta(x^6 + A) / eta(x^3 + A))^9, n))};
(Magma) M := Basis(ModularForms(Gamma1(6), 1), 43); M1 := M[1]; M2 := M[2]; A<q> := M2/(M1 + 2*M2); A; /* Michael Somos, May 26 2023 */
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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