A128640 Expansion of (1/3) * (c(q^2)^2 / c(q)) / (b(q^2)^2 / b(q)) in powers of q where b(), c() are cubic AGM theta functions.

1, -4, 10, -20, 39, -76, 140, -244, 415, -696, 1140, -1820, 2861, -4448, 6816, -10292, 15372, -22756, 33356, -48408, 69683, -99600, 141312, -199036, 278557, -387608, 536230, -737632, 1009464, -1374888, 1863764, -2514868, 3378948, -4521672, 6027000, -8002676

Offset

1,2

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

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q * (psi(q^3) / psi(q))^4 in powers of q where psi() is a Ramanujan theta function.

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

Euler transform of period 6 sequence [ -4, 4, 0, 4, -4, 0, ...].

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = v * (1-u) * (1-9*u) - (u-v)^2.

G.f.: x * (Product_{k>0} (1 - x^k + x^(2*k))^2 * (1 + x^k + x^(2*k)) )^4.

G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (81*u^2*v^2 + 9*u*v - 12*u + 30*u^2 - 108*u^2*v + 1) * v - u^3.

G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = (1/9) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A128637.

a(n) = - A128641(n) unless n = 0. Convolution inverse of A128633.

a(n) ~ -(-1)^n * exp(2*Pi*sqrt(n/3)) / (2 * 3^(9/4) * n^(3/4)). - Vaclav Kotesovec, Jun 06 2018

Empirical: Sum_{n>=1} a(n)/exp(2*Pi*n) = 1/3 + (2/9)*sqrt(3) - (2/9)*sqrt(6)*3^(1/4). - Simon Plouffe, Mar 02 2021

Example

G.f. = q - 4*q^2 + 10*q^3 - 20*q^4 + 39*q^5 - 76*q^6 + 140*q^7 - 244*q^8 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, q^(3/2)] / EllipticTheta[ 2, 0, q^(1/2)])^4, {q, 0, n}]; (* Michael Somos, Aug 26 2015 *)

Prog

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

Crossrefs

Cf. A128633, A128641, A164617.

Sequence in context: A090164 A261635 A258346 * A128641 A164617 A366646

Adjacent sequences: A128637 A128638 A128639 * A128641 A128642 A128643

Keyword

sign

Author

Michael Somos, Mar 16 2007

Status

approved