A260215 Expansion of chi(-q) * chi(q^9) / (chi(q) * chi(-q^9)) in powers of q where chi() is a Ramanujan theta function.

1, -2, 2, -4, 6, -8, 12, -16, 22, -28, 36, -48, 60, -76, 96, -120, 150, -184, 228, -280, 340, -416, 504, -608, 732, -878, 1052, -1252, 1488, -1768, 2088, -2464, 2902, -3408, 3996, -4672, 5460, -6364, 7400, -8600, 9972, -11544, 13344, -15400, 17752, -20424

Offset

0,2

Comments

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

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of psi(-q) * psi(q^9) / (psi(q) * psi(-q^9)) in powers of q where psi() is a Ramanujan theta function.

Expansion of eta(q)^2 * eta(q^4) * eta(q^18)^3 / (eta(q^2)^3 * eta(q^9)^2 * eta(q^36)) in powers of q.

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

G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = g(t) where q = exp(2 Pi i t) and g() is the g.f. of A128143.

a(n) = (-1)^n * A261156(n). Convolution inverse of A261156

a(2*n + 1) = -2 * A261203(n) = -2 * A261154(2*n + 1). 2 * a(2*n) = A261154(2*n) unless n=0.

a(3*n) = A261320(n). a(3*n + 1) = -2 * A261325(n). a(3*n + 2) = 2 * A260057(n). - Michael Somos, Nov 08 2015

a(n) ~ (-1)^n * exp(2*Pi*sqrt(n)/3) / (2*sqrt(3)*n^(3/4)). - Vaclav Kotesovec, Nov 16 2017

Example

G.f. = 1 - 2*x + 2*x^2 - 4*x^3 + 6*x^4 - 8*x^5 + 12*x^6 - 16*x^7 + 22*x^8 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ q, q^2] QPochhammer[ q, -q] QPochhammer[ -q^9, q^18] QPochhammer[ -q^9, q^9], {q, 0, n}];

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A) * eta(x^18 + A)^3 / (eta(x^2 + A)^3 * eta(x^9 + A)^2 * eta(x^36 + A)), n))};

Crossrefs

Cf. A128143, A260057, A261154, A261156, A261203, A261230, A261325.

Sequence in context: A329899 A051466 A320193 * A261156 A080015 A210030

Adjacent sequences: A260212 A260213 A260214 * A260216 A260217 A260218

Keyword

sign

Author

Michael Somos, Aug 13 2015

Status

approved