A133573 Expansion of ( 5 * phi(-q^5)^2 - phi(-q)^2 ) / 4 in powers of q where phi() is a Ramanujan theta function.

1, 1, -1, 0, -1, -3, 0, 0, -1, 1, 3, 0, 0, 2, 0, 0, -1, 2, -1, 0, 3, 0, 0, 0, 0, -7, -2, 0, 0, 2, 0, 0, -1, 0, -2, 0, -1, 2, 0, 0, 3, 2, 0, 0, 0, -3, 0, 0, 0, 1, 7, 0, -2, 2, 0, 0, 0, 0, -2, 0, 0, 2, 0, 0, -1, -6, 0, 0, -2, 0, 0, 0, -1, 2, -2, 0, 0, 0, 0, 0, 3, 1, -2, 0, 0, -6, 0, 0, 0, 2, 3, 0, 0, 0, 0, 0, 0, 2, -1, 0, 7, 2, 0, 0, -2

Offset: 0

Author

Michael Somos, Sep 17 2007

Keywords

• sign
• mult
• changed

Comments

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

Multiplicative because this sequence is the inverse Moebius transform of a multiplicative sequence. - Andrew Howroyd, Aug 06 2018

Examples

			G.f. = 1 + q - q^2 - q^4 - 3*q^5 - q^8 + q^9 + 3*q^10 + 2*q^13 - q^16 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Shaun Cooper, On Ramanujan's function k(q)=r(q)r^2(q^2), Ramanujan J., 20 (2009), 311-328; see p. 313, eq. (2.5).
• Li-Chien Shen, On the additive formulas of the theta functions and a collection of Lambert series pertaining to the modular equations of degree 5, Trans. Amer. Math. Soc. 345 (1994), no. 1, 323-345; see p. 335, eq. (3.5), p. 342, eq. (3.42).
• Michael Somos, Introduction to Ramanujan theta functions, 2019.
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Crossrefs

Cf. A122190, A133574, A214316.

Cf. A000700, A000122, A010054, A121373.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ (5 EllipticTheta[ 4, 0, q^5]^2 - EllipticTheta[ 4, 0, q]^2)/4, {q, 0, n}]; (* Michael Somos, Jul 12 2012 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ q^2]^2 QPochhammer[ q^5, q^10] / QPochhammer[ q, q^2], {q, 0, n}]; (* Michael Somos, Jul 12 2012 *)

PARI

{a(n) = if( n<1, n==0, (-1)^n * sumdiv(n, d, if( d%5==0, kronecker(-4, d/5) * 5) - kronecker(-4, d)))};

PARI

{a(n) = local(A); if( n<0, 0, A = x*O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^5+A) / (eta(x + A) * eta(x^10 + A)), n))};

Formula

Expansion of eta(q^2)^3 * eta(q^5) / ( eta(q) * eta(q^10) ) in powers of q.

Euler transform of period 10 sequence [ 1, -2, 1, -2, 0, -2, 1, -2, 1, -2, ...].

Moebius transform is period 40 sequence [ 1, -2, -1, 0, -4, 2, -1, 0, 1, 8, -1, 0, 1, 2, 4, 0, 1, -2, -1, 0, 1, 2, -1, 0, -4, -2, -1, 0, 1, -8, -1, 0, 1, -2, 4, 0, 1, 2, -1, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (40 t)) = 20 (t/i) g(t) where q = exp(2 Pi i t) and g() is g.f. for A122190.

a(n) = (-1)^n * A133574(n). a(2*n) = A133574(n). a(4*n + 1) = A214316(n). a(4*n + 3) = a(9*n + 3) = a(9*n + 6) = 0. a(9*n) = a(n). - Michael Somos, Jul 12 2012

Sum_{k=1..n} abs(a(k)) ~ (8*Pi/25) * n. - Amiram Eldar, Jan 27 2024