A261988 Expansion of phi(q^9) / phi(q) in powers of q where phi() is a Ramanujan theta function.

1, -2, 4, -8, 14, -24, 40, -64, 100, -152, 228, -336, 488, -700, 992, -1392, 1934, -2664, 3640, -4936, 6648, -8896, 11832, -15648, 20584, -26942, 35096, -45512, 58768, -75576, 96816, -123568, 157156, -199200, 251676, -316992, 398072, -498460, 622448, -775216

Offset

0,2

Comments

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

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

G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = (1/3) g(t) where q = ellq(2 Pi i t) and g() is the g.f. for A139380.

a(n) = (-1)^n * A128770(n). Convolution inverse is A139380.

Example

G.f. = 1 - 2*q + 4*q^2 - 8*q^3 + 14*q^4 - 24*q^5 + 40*q^6 - 64*q^7 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^9] / EllipticTheta[ 3, 0, q], {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)^2 * eta(x^18 + A)^5 / (eta(x^2 + A)^5 * eta(x^9 + A)^2 * eta(x^36 + A)^2), n))};

Crossrefs

Cf. A128770, A139380.

Sequence in context: A365666 A090399 A069251 * A128770 A280947 A069252

Adjacent sequences: A261985 A261986 A261987 * A261989 A261990 A261991

Keyword

sign

Author

Michael Somos, Sep 07 2015

Status

approved