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A128636
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Expansion of 3 * (b(q^2)^2 / b(q)) / (c(q)^2 / c(q^2)) in powers of q where b(), c() are cubic AGM theta functions.
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4
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1, 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
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OFFSET
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0,3
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COMMENTS
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LINKS
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FORMULA
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Expansion of (psi(q)^3 / psi(q^3)) / (phi(-q^3)^3 / phi(-q)) in powers of q where phi(), psi() are Ramanujan theta functions.
Expansion of (eta(q^6) / eta(q)) * (eta(q^2) / eta(q^3))^5 in powers of q.
Euler transform of period 6 sequence [ 1, -4, 6, -4, 1, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = v* (1-v)* (9-8*u) + (u-v)^2.
G.f.: Product_{k>0} (1 - x^(6*k)) / (1 - x^k) * ((1 - x^(2*k)) / (1 - x^(3*k)))^5.
Empirical: Sum_{n>=0} a(n)/exp(2*Pi*n) = -(3/8)*sqrt(3) + (3/8)*sqrt(9 + 6*sqrt(3)). - Simon Plouffe, Mar 02 2021
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EXAMPLE
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G.f. = 1 + q - 3*q^2 + 3*q^3 + 5*q^4 - 18*q^5 + 15*q^6 + 24*q^7 - 75*q^8 + ...
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MATHEMATICA
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eta[x_] := x^(1/24)*QPochhammer[x]; A128636[n_] := SeriesCoefficient[(eta[q^6]/eta[q])*(eta[q^2]/eta[q^3])^5, {q, 0, n}]; Table[A128636[n], {n, 0, 50}] (* G. C. Greubel, Aug 21 2017 *)
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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^2 + A) / eta(x^3 + A))^5 * eta(x^6 + A) / eta(x + A), n))};
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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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