A144377 Expansion of phi(q) / phi(q^5) in powers of q where phi() is a Ramanujan theta function.
1, 2, 0, 0, 2, -2, -4, 0, 0, -2, 4, 8, 0, 0, 4, -8, -14, 0, 0, -8, 14, 24, 0, 0, 12, -22, -40, 0, 0, -20, 36, 64, 0, 0, 32, -56, -98, 0, 0, -48, 84, 148, 0, 0, 72, -126, -220, 0, 0, -106, 184, 320, 0, 0, 152, -264, -460, 0, 0, -216, 376, 652, 0, 0, 306, -528
Offset: 0
Keywords
Examples
G.f. = 1 + 2*q + 2*q^4 - 2*q^5 - 4*q^6 - 2*q^9 + 4*q^10 + 8*q^11 + 4*q^14 + ...
References
- B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 235, Entry 67.
Links
- Michael Somos, Introduction to Ramanujan theta functions
- Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Programs
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Mathematica
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] / EllipticTheta[ 3, 0, q^5], {q, 0, n}]; (* Michael Somos, Sep 06 2015 *) -
PARI
{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^5 + A)^2 * eta(x^20 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^10 + A)^5), n))};
Formula
Expansion of eta(q^2)^5 * eta(q^5)^2 * eta(q^20)^2 / (eta(q)^2 * eta(q^4)^2 * eta(q^10)^5) in powers of q.
Euler transform of period 20 sequence [ 2, -3, 2, -1, 0, -3, 2, -1, 2, 0, 2, -1, 2, -3, 0, -1, 2, -3, 2, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u^4 - 2*u^2 +5) * (v^4 - 2*v^2 + 5) - 4 * (u^2 - 2*u*v - v^2)^2.
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (v^2 + 3*u*v - u^2) * (u^2 + v^2) - u*v * (5 + u^2*v^2).
G.f. is a period 1 Fourier series which satisfies f(-1 / (20 t)) = 5^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A261968.
G.f.: Product_{k>0} P(20, x^k)^2 / (P(10, x^k)^3 * P(5, x^k)) where P(n, x) is the n-th cyclotomic polynomial.
a(5*n + 2) = a(5*n + 3) = 0.
Comments