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A138559
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Expansion of phi(x) * chi(-x) in powers of x where phi(), chi() are Ramanujan theta functions.
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2
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1, 1, -2, -1, 1, -1, -1, -1, 2, 2, -2, 0, 1, 1, -1, 0, 3, 1, -3, -2, 3, 0, -2, -1, 3, 2, -4, -2, 2, 1, -4, -2, 5, 3, -6, -1, 5, 1, -5, -3, 6, 3, -6, -3, 7, 2, -6, -2, 9, 5, -10, -5, 9, 3, -9, -4, 11, 6, -12, -4, 11, 5, -12, -5, 14, 6, -16, -7, 15, 5, -16, -7, 19, 9, -20, -8, 19, 7, -20, -10, 24, 11, -25, -11, 24, 9, -26, -11, 29, 13, -31, -13
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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 phi(x) * chi(-x) = phi(-x^2) * chi(x) = psi(x) * chi(-x^2)^2 = f(-x) * chi(x)^2 = f(x) * chi(-x^2) = f(x)^2 / psi(x) = phi(-x^2)^2 / f(-x) if powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions.
Expansion of q^(1/24) * eta(q^2)^4 / (eta(q^4)^2 * eta(q)) in powers of q.
Euler transform of period 4 sequence [ 1, -3, 1, -1, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (2304 t)) = 96^(1/2) (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A096920.
G.f.: Product_{k>0} (1 - x^k) * (1 + x^(2*k-1))^2.
a(n) ~ c1(n) * exp(Pi*sqrt(2*(n - 1/24)/3)/4) / (2*sqrt(2*(n - 1/24))), where c1(n) = (-1)^(n/2) * 1.847759... if n is even and c1(n) = -(-1)^((n+1)/2) * 0.765366... if n is odd [Chan, 2005, formula 2.2 and 2.7, where a1(n) = a(n) if n is even and a1(n) = -a(n) if n is odd]. - Vaclav Kotesovec, May 08 2020
In closed form, abs(c1(n)) = sqrt(2 + (-1)^n*sqrt(2)). - Vaclav Kotesovec, May 08 2020
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EXAMPLE
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G.f. = 1 + x - 2*x^2 - x^3 + x^4 - x^5 x^6 - x^7 + 2*x^8 + 2*x^9 - 2*x^10 + ...
G.f. = 1/q + q^23 - 2*q^47 - q^71 + q^95 - q^119 - q^143 - q^167 + 2*q^191 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] QPochhammer[ x, x^2], {x, 0, n}]; (* Michael Somos, Aug 31 2014 *)
CoefficientList[Series[QPochhammer[-x] * QPochhammer[x^2] / QPochhammer[x^4], {x, 0, 100}], x] (* Vaclav Kotesovec, May 08 2020 *)
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PROG
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(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^4 / (eta(x + A) * eta(x^4 + A)^2), 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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