A139139 Expansion of (phi(q) / phi(q^3) - 1) / 2 in powers of q where phi() is a Ramanujan theta function.

1, 0, -1, -1, 0, 2, 2, 0, -3, -4, 0, 5, 6, 0, -8, -9, 0, 12, 14, 0, -18, -20, 0, 26, 29, 0, -37, -42, 0, 52, 58, 0, -72, -80, 0, 99, 110, 0, -134, -148, 0, 180, 198, 0, -240, -264, 0, 317, 347, 0, -416, -454, 0, 542, 592, 0, -702, -764, 0, 904, 982, 0, -1158

Offset

1,6

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 = 1..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q * f(-q, -q^11) / f(-q, q^2) in powers of q where f(, ) is the Ramanujan general theta function. - Michael Somos, Sep 07 2015

Expansion of q * chi(-q^2) * psi(q^6)^2 / (psi(q^3) * f(-q^5, -q^7)) in powers of q where phi(), f() are Ramanujan theta functions.

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

G.f.: ((Sum_{k in Z} x^k^2) / (Sum_{k in Z} x^(3*k^2)) - 1) / 2.

G.f.: Product_{k>0} (1 + x^(2*k))^2 * (1 - x^(2*k) + x^(4*k))^3 / ( (1 + x^k) * (1 - x^k + x^(2*k)) * (1 - x^(12*k - 5)) * (1 - x^(12*k - 7))).

2 * a(n) = A139137(n) unless n=0. a(3*n + 2) = 0.

a(3*n + 1) = A139135(n). - Michael Somos, Sep 07 2015

Example

G.f. = q - q^3 - q^4 + 2*q^6 + 2*q^7 - 3*q^9 - 4*q^10 + 5*q^12 + 6*q^13 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] / EllipticTheta[ 3, 0, q^3] - 1) / 2, {q, 0, n}]; (* Michael Somos, Sep 07 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^5 * eta(x^3 + A)^2 * eta(x^12 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^6 + A)^5) - 1) / 2, n))};

Crossrefs

Cf. A139135, A139137.

Sequence in context: A207383 A191362 A137422 * A077872 A300453 A239292

Adjacent sequences: A139136 A139137 A139138 * A139140 A139141 A139142

Keyword

sign

Author

Michael Somos, Apr 10 2008

Status

approved