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A123649
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Expansion of c(q^4) / c(q) in powers of q where c() is a cubic AGM theta function.
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5
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1, -1, -1, 3, -2, -3, 8, -5, -7, 18, -12, -15, 38, -24, -30, 75, -46, -57, 140, -86, -104, 252, -152, -183, 439, -262, -313, 744, -442, -522, 1232, -725, -852, 1998, -1168, -1365, 3182, -1852, -2150, 4986, -2886, -3336, 7700, -4436, -5106, 11736, -6736, -7719, 17673, -10103, -11538, 26322
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OFFSET
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1,4
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COMMENTS
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Empirical: Sum_{n>=1} a(n)/exp(Pi*n) = 7/4 + (3/4)*sqrt(3) - (1/4)*sqrt(72 + 42*sqrt(3)). - Simon Plouffe, Mar 01 2021
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LINKS
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FORMULA
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Expansion of q * psi(q^6) * f(q, q^5) / f(q, q^2)^2 = q * f(-q^2, -q^10)^2 / (phi(-q^3) * f(q, q^5)) in powers of q where psi(), phi(), f() are Ramanujan theta functions. - Michael Somos, Jul 23 2013
Expansion of q * psi(-q) * psi(q^6)^3 / (psi(q^2) * psi(-q^3)^3) in powers of q where psi() is a Ramanujan theta function. - Michael Somos, Jul 23 2013
Expansion of eta(q) * eta(q^12)^3 / (eta(q^4) * eta(q^3)^3) in powers of q.
Euler transform of period 12 sequence [ -1, -1, 2, 0, -1, 2, -1, 0, 2, -1, -1, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 - v*(1 - 2*u) * (1 - 2*v).
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EXAMPLE
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q - q^2 - q^3 + 3*q^4 - 2*q^5 - 3*q^6 + 8*q^7 - 5*q^8 - 7*q^9 + ...
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MATHEMATICA
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A123649[n_] := SeriesCoefficient[q*QPochhammer[q] *QPochhammer[q^12]^3/ (QPochhammer[q^4]* QPochhammer[q^3]^3), {q, 0, n}]; Rest[Table[ A123649[n], {n, 0, 50}]] (* G. C. Greubel, Oct 17 2017 *)
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PROG
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(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^12 + A)^3 / eta(x^3 + A)^3 / eta(x^4 + 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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