A145708 Expansion of psi(-q) / psi(-q^5) in powers of q where psi() is a Ramanujan theta function.

1, -1, 0, -1, 0, 1, 0, 0, -1, 0, 2, 0, 0, -1, 0, 2, -1, 0, -2, 0, 3, -2, 0, -3, 0, 5, -2, 0, -3, 0, 6, -2, 0, -4, 0, 8, -3, 0, -6, 0, 11, -5, 0, -8, 0, 14, -6, 0, -10, 0, 18, -6, 0, -12, 0, 22, -9, 0, -16, 0, 28, -13, 0, -21, 0, 36, -14, 0, -25, 0, 44, -16, 0

Offset: 0

Author

Michael Somos, Oct 17 2008

Keywords

• sign

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = 1 - x - x^3 + x^5 - x^8 + 2*x^10 - x^13 + 2*x^15 - x^16 - 2*x^18 + ...
G.f. = 1/q - q - q^5 + q^9 - q^15 + 2*q^19 - q^25 + 2*q^29 - q^31 - 2*q^35 + ...
		

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

Crossrefs

Cf. A036026, A138532, A145723, A145740, A146164.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ x^(1/2) EllipticTheta[ 2, Pi/4, x^(1/2)] / EllipticTheta[ 2, Pi/4, x^(5/2)], {x, 0, n}]; (* Michael Somos, Sep 06 2015 *)

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^10 + A) / (eta(x^2 + A) * eta(x^5 + A) * eta(x^20 + A)), n))};

Formula

Expansion of q^(1/2) * eta(q) * eta(q^4) * eta(q^10) / (eta(q^2) * eta(q^5) * eta(q^20)) in powers of q.

Euler transform of period 20 sequence [ -1, 0, -1, -1, 0, 0, -1, -1, -1, 0, -1, -1, -1, 0, 0, -1, -1, 0, -1, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (80 t)) = 5^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A036026.

a(5*n + 2) = a(5*n + 4) = 0.

G.f.: (Product_{k>0} P(5, x^k) * P(20, x^k))^(-1) where P(n, x) is the n-th cyclotomic polynomial.

a(n) = (-1)^n * A138532(n). a(5*n + 3) = - A036026(n).

Convolution square is A145740. Convolution inverse is A036026.

a(n) = A145723(2*n - 1). a(2*n) = A146164(n). a(2*n + 1) = - A147699(n). - Michael Somos, Sep 06 2015