A131124 Expansion of q^(-1) * (phi(-q) / psi(q^4))^2 in powers of q where phi(), psi() are Ramanujan theta functions.

1, -4, 4, 0, 2, 0, -8, 0, -1, 0, 20, 0, -2, 0, -40, 0, 3, 0, 72, 0, 2, 0, -128, 0, -4, 0, 220, 0, -4, 0, -360, 0, 5, 0, 576, 0, 8, 0, -904, 0, -8, 0, 1384, 0, -10, 0, -2088, 0, 11, 0, 3108, 0, 12, 0, -4552, 0, -15, 0, 6592, 0, -18, 0, -9448, 0, 22, 0, 13392

Offset

-1,2

Comments

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

Number 3 of the 15 generalized eta-quotients listed in Table I of Yang 2004. - Michael Somos, Jul 21 2014

A generator (Hauptmodul) of the function field associated with congruence subgroup Gamma_0(8). [Yang 2004] - Michael Somos, Jul 21 2014

Links

G. C. Greubel, Table of n, a(n) for n = -1..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Y. Yang, Transformation formulas for generalized Dedekind eta functions, Bull. London Math. Soc. 36 (2004), no. 5, 671-682. See p. 679, Table 1.

Formula

Expansion of (eta(q)^2 * eta(q^4) / (eta(q^2) * eta(q^8)^2 ))^2 in powers of q.

Euler transform of period 8 sequence [ -4, -2, -4, -4, -4, -2, -4, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 32 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A107035.

G.f. A(q) satisfies 0 = f(A(q), A(q^2)) where f(u, v) = u * (u + 8) * (v + 4) - v^2.

G.f.: (x * Product_{k>0} (1 + x^k)^4 * (1 + x^(2*k))^2 * (1 + x^(4*k))^4 )^-1.

Convolution inverse of A107035.

a(2*n) = 0 unless n=0. a(n) = A029841(n) unless n=0. a(4*n - 1) = A029839(n). a(4*n + 1) = 4 * A079006(n).

Example

G.f. = 1/q - 4 + 4*q + 2*q^3 - 8*q^5 - q^7 + 20*q^9 - 2*q^11 - 40*q^13 + 3*q^15 + ...

Mathematica

a[ n_] := SeriesCoefficient[ 4 (EllipticTheta[ 4, 0, q] / EllipticTheta[ 2, 0, q^2])^2, {q, 0, n}]; (* Michael Somos, Apr 24 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x + A)^2 * eta(x^4 + A) / (eta(x^2 + A) * eta(x^8 + A)^2) )^2, n))};

Crossrefs

Cf. A029839, A029841, A079006, A107035.

Sequence in context: A108068 A283998 A318143 * A131125 A187149 A106508

Adjacent sequences: A131121 A131122 A131123 * A131125 A131126 A131127

Keyword

sign

Author

Michael Somos, Jun 15 2007

Status

approved