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A138526
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Expansion of phi(-q^5) / phi(-q) in powers of q where phi() is a Ramanujan theta function.
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11
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1, 2, 4, 8, 14, 22, 36, 56, 84, 126, 184, 264, 376, 528, 732, 1008, 1374, 1856, 2492, 3320, 4394, 5784, 7568, 9848, 12756, 16442, 21096, 26960, 34312, 43500, 54956, 69184, 86804, 108576, 135392, 168336, 208722, 258096, 318320, 391632, 480664
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OFFSET
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0,2
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COMMENTS
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LINKS
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FORMULA
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Expansion of (eta(q^5) / eta(q))^2 * eta(q^2) / eta(q^10) in powers of q.
Euler transform of period 10 sequence [ 2, 1, 2, 1, 0, 1, 2, 1, 2, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u^2 - v^2)^2 - u^2 * (v^2 - 1) * (5*v^2 - 1).
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (u^2 - v^2) * (u + v)^2 - u * v * (u^2 - 1) * (5*v^2 - 1).
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = (u - v)^2 * w^2 - u * v * (v^2 - 1).
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 * u6 - u2 * u3)^2 - u1 * u3 * (u2^2 - u6^2).
G.f. is a period 1 Fourier series which satisfies f(-1 / (10 t)) = 5^(-1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A138532.
G.f.: Product_{k>0} P(5, x^k) / P(10, x^k) where P(n, x) is the n-th cyclotomic polynomial.
a(n) ~ exp(2*Pi*sqrt(n/5)) / (2*5^(3/4)*n^(3/4)). - Vaclav Kotesovec, Sep 01 2015
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EXAMPLE
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G.f. = 1 + 2*q + 4*q^2 + 8*q^3 + 14*q^4 + 22*q^5 + 36*q^6 + 56*q^7 + 84*q^8 + ...
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MATHEMATICA
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nmax=50; CoefficientList[Series[Product[(1+x^k)*(1-x^(5*k))/((1-x^k)*(1+x^(5*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 01 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q^5] / EllipticTheta[ 4, 0, q], {q, 0, n}]; (* Michael Somos, Sep 14 2015 *)
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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^5 + A) / eta(x + A))^2 * eta(x^2 + A) / eta(x^10 + A), 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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STATUS
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approved
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