A134079 Expansion of q^(-2/3) * c(-q)^2 / 9 in powers of q where c(q) is a cubic AGM theta function.

1, -2, 5, -4, 8, -6, 14, -8, 14, -10, 21, -16, 20, -14, 28, -16, 31, -18, 40, -20, 32, -28, 42, -24, 38, -32, 62, -28, 44, -30, 56, -40, 57, -34, 70, -36, 72, -38, 70, -48, 62, -52, 85, -44, 68, -46, 112, -56, 74, -50, 100, -64, 80, -64, 98, -56, 108, -58, 124

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..10000

Michael Somos, Introduction to Ramanujan theta functions, 2019.

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

Formula

Expansion of ( f(x^3)^3 / f(x) )^2 in powers of x where f() is a Ramanujan theta function.

Expansion of q^(-2/3) * eta(q)^2 * eta(q^4)^2 * eta(q^6)^18 / (eta(q^2) * eta(q^3)* eta(q^12))^6 in powers of q.

Euler transform of period 12 sequence [ -2, 4, 4, 2, -2, -8, -2, 2, 4, 4, -2, -4, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = (4/3) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A263773.

a(n) = (-1)^n * A033686(n). 18 * a(n) = A134078(3*n + 2).

From Michael Somos, Feb 19 2015: (Start)

a(2*n + 1) = -2 * A098098(n).

Convolution square of A227696. (End)

Sum_{k=1..n} a(k) ~ (Pi^2/54) * n^2. - Amiram Eldar, Nov 23 2023

Example

G.f. = 1 - 2*x + 5*x^2 - 4*x^3 + 8*x^4 - 6*x^5 + 14*x^6 - 8*x^7 + 14*x^8 - ...
G.f. = q^2 - 2*q^5 + 5*q^8 - 4*q^11 + 8*q^14 - 6*q^17 + 14*q^20 - 8*q^23 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (QPochhammer[ -x^3]^3 / QPochhammer[ -x])^2, {x, 0, n}]; (* Michael Somos, Feb 19 2015 *)
a[ n_] := If[ n < 0, 0, (-1)^n DivisorSigma[ 1, 3 n + 2] / 3]; (* Michael Somos, Feb 19 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x*O(x^n); polcoeff( ( eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^9 / ( eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A) )^3 )^2, n))};
(PARI) {a(n) = if( n<0, 0, (-1)^n * sigma(3*n + 2) / 3)}; /* Michael Somos, Feb 19 2015 */

Crossrefs

Cf. A033686, A098098, A134078, A227696, A263773.

Cf. A000700, A000122, A004016, A005882, A005928, A010054, A121373.

Sequence in context: A093052 A081556 A187012 * A033686 A243973 A286015

Adjacent sequences: A134076 A134077 A134078 * A134080 A134081 A134082

Keyword

sign

Author

Michael Somos, Oct 06 2007

Status

approved