A138526 Expansion of phi(-q^5) / phi(-q) in powers of q where phi() is a Ramanujan theta function.
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
Offset: 0
Keywords
Examples
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 + ...
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Michael Somos, Introduction to Ramanujan theta functions
- Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Programs
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Mathematica
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 *) -
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))};
Formula
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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