A128111 Expansion of q^(-1) * (phi(q) / phi(q^9) - 1) / 2 in powers of q^3 where phi() is a Ramanujan theta function.

1, 1, 0, -2, -2, 1, 4, 4, -1, -8, -8, 2, 14, 14, -4, -24, -23, 6, 40, 38, -10, -63, -60, 16, 98, 92, -24, -150, -140, 36, 224, 208, -54, -329, -304, 78, 478, 440, -112, -684, -627, 160, 968, 884, -224, -1358, -1236, 312, 1884, 1710, -432, -2592, -2346, 590, 3540, 3196, -801, -4796, -4320, 1082, 6454

Offset

0,4

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).

References

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 354 Eq. (3.11)

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

Expansion of -q^(-1/3) * c(q^2) / c(-q) in powers of q where c() is a cubic AGM theta function.

Expansion of q^(-1/3) * (eta(q^2)^2 * eta(q^3)^3 * eta(q^12)^3) / (eta(q) * eta(q^4) * eta(q^6)^6) in powers of q.

Given g.f. A(x), then B(q) = q*A(q^3) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (1 - u^3) * (1 - v^3) - (1 + 2*u*v) * (1 - u*v)^3.

Given g.f. A(x), then B(q) = q*A(q^3) satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (v + v^2 + v^3) - u^3*(1 - 2*v + 4*v^2).

Given g.f. A(x), then B(q) = q*A(q^3) satisfies 0 = f(B(q), B(q^7)) where f(u, v) = (u-v)^8 - u*(1 - u^3) * (1 + 8*u^3) * v * (1 - v^3) * (1 + 8*v^3).

Given g.f. A(x), then B(q) = q*A(q^3) satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = (v^2 - u) * (v^2 - w) - v^2*(v - u^2) * (v - w^2).

Given g.f. A(x), then B(q) = q*A(q^3) satisfies 0 = f(B(q), B(q^2), B(q^3), B(q^6)) where f(u1, u2, u3, u6) = u6 + u3 - u1*u2 + u6*u3 * (1 - 2*u1*u2).

Euler transform of period 12 sequence [ 1, -1, -2, 0, 1, 2, 1, 0, -2, -1, 1, 0, ...].

(-1)^n * a(n) = A092848(n).

Convolution inverse of A062244.

Example

G.f. = 1 + x - 2*x^3 - 2*x^4 + x^5 + 4*x^6 + 4*x^7 - x^8 - 8*x^9 - 8*x^10 + ...
G.f. = q + q^4 - 2*q^10 - 2*q^13 + q^16 + 4*q^19 + 4*q^22 - q^25 - 8*q^28 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] / QPochhammer[ -x^3, x^6]^3, {x, 0, n}]; (* Michael Somos, Feb 19 2015 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, x] / EllipticTheta[ 3, 0, x^9] - 1) / (2 x), {x, 0, 3 n}]; (* Michael Somos, Feb 19 2015 *)

Prog

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A)^3 * eta(x^12 + A)^3 / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^6), n))};

Crossrefs

Cf. A062244, A092848.

Sequence in context: A110090 A196831 A092848 * A107356 A329854 A124725

Adjacent sequences: A128108 A128109 A128110 * A128112 A128113 A128114

Keyword

sign

Author

Michael Somos, Feb 14 2007

Status

approved