A186115 Expansion of q^(-1) * f(-q^3) * phi(-q^3) / (phi(-q^2) * psi(-q^9)) in powers of q where f(), phi(), psi() are Ramanujan theta functions.

1, 0, 2, -3, 4, -6, 9, -12, 16, -21, 28, -36, 47, -60, 76, -96, 120, -150, 185, -228, 280, -342, 416, -504, 608, -732, 878, -1050, 1252, -1488, 1765, -2088, 2464, -2901, 3408, -3996, 4676, -5460, 6364, -7404, 8600, -9972, 11545, -13344, 15400, -17748, 20424

Offset

-1,3

Comments

Denoted by "(36~q)" in Simon Norton's replicable function list.

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 = -1..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Formula

Expansion of eta(q^3)^3 * eta(q^4) * eta(q^18) / (eta(q^2)^2 * eta(q^6) * eta(q^9) * eta(q^36)) in powers of q.

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

G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A261154. - Michael Somos, Aug 10 2015

a(n) = -(-1)^n * A058647(n).

a(2*n) = -3 * A128129(n). a(3*n) = 4 * A228447(n). - Michael Somos, Aug 10 2015

Example

G.f. = 1/q + 2*q - 3*q^2 + 4*q^3 - 6*q^4 + 9*q^5 - 12*q^6 + 16*q^7 + ...

Mathematica

a[ n_] := SeriesCoefficient[ 2 q^(1/8) QPochhammer[ q^3] EllipticTheta[ 4, 0, q^3] / (EllipticTheta[ 4, 0, q^2] EllipticTheta[ 2, 0, q^(9/2)]), {q, 0, n}]; (* Michael Somos, Aug 10 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x^3 + A)^3 * eta(x^4 + A) * eta(x^18 + A) / (eta(x^2 + A)^2 * eta(x^6 + A) * eta(x^9 + A) * eta(x^36 + A)), n))};

Crossrefs

Cf. A058647, A228447, A261154.

Sequence in context: A229169 A155510 A073576 * A187020 A058647 A305630

Adjacent sequences: A186112 A186113 A186114 * A186116 A186117 A186118

Keyword

sign

Author

Michael Somos, Feb 13 2011

Status

approved