A139380 Expansion of phi(q) / phi(q^9) in powers of q where phi() is a Ramanujan theta function.

1, 2, 0, 0, 2, 0, 0, 0, 0, 0, -4, 0, 0, -4, 0, 0, 2, 0, 0, 8, 0, 0, 8, 0, 0, -2, 0, 0, -16, 0, 0, -16, 0, 0, 4, 0, 0, 28, 0, 0, 28, 0, 0, -8, 0, 0, -48, 0, 0, -46, 0, 0, 12, 0, 0, 80, 0, 0, 76, 0, 0, -20, 0, 0, -126, 0, 0, -120, 0, 0, 32, 0, 0, 196, 0, 0, 184

Offset

0,2

Comments

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

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

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

Formula

Expansion of 1 + 2 * q * chi(q^3) / chi(q^9)^3 in powers of q where chi() is a Ramanujan theta function.

Expansion of 1 - 2 * c(q^6) / c(-q^3) in powers of q where c() is a cubic AGM theta function.

Expansion of eta(q^2)^5 * eta(q^9)^2 * eta(q^36)^2 / (eta(q)^2 * eta(q^4)^2 * eta(q^18)^5) in powers of q.

Euler transform of period 36 sequence [ 2, -3, 2, -1, 2, -3, 2, -1, 0, -3, 2, -1, 2, -3, 2, -1, 2, 0, 2, -1, 2, -3, 2, -1, 2, -3, 0, -1, 2, -3, 2, -1, 2, -3, 2, 0, ...].

G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (u - v)^3 - u * (3 - u) * (v - 1) * (3 - 2*u + u*v).

G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A261988. - Michael Somos, Sep 07 2015

G.f.: (1 + 2 * Sum_{k>0} x^k^2) / (1 + 2 * Sum_{k>0} x^(9*k^2)).

G.f.: Product_{k>0} (1 - x^(2*k)) * (1 + x^(2*k-1))^2 / ((1 - x^(18*k)) * (1 + x^(18*k-9))^2).

a(n) = A128771(n) unless n=0. a(n) = (-1)^n * A128771(n).

a(3*n) = 0 unless n=0. a(3*n + 2) = 0. a(3*n + 1) = 2 * A128111(n).

Empirical : sum(exp(-Pi/3)^(n-1)*a(n),n=1..infinity) = sqrt(3). Simon Plouffe, Feb. 20, 2011.

Convolution inverse is A261988. - Michael Somos, Sep 07 2015

Example

G.f. = 1 + 2*q + 2*q^4 - 4*q^10 - 4*q^13 + 2*q^16 + 8*q^19 + 8*q^22 - 2*q^25 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] / EllipticTheta[ 3, 0, q^9], {q, 0, n}]; (* Michael Somos, Sep 07 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^9 + A)^2 * eta(x^36 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^18 + A)^5), n))};

Crossrefs

Cf. A128111, A128771, A261988.

Sequence in context: A249772 A193531 A093492 * A128771 A000122 A002448

Adjacent sequences: A139377 A139378 A139379 * A139381 A139382 A139383

Keyword

sign

Author

Michael Somos, Apr 15 2008

Status

approved