A137830 Expansion of phi(-x) / f(-x^4)^2 in powers of x where phi(), f() are Ramanujan theta functions.

1, -2, 0, 0, 4, -4, 0, 0, 9, -12, 0, 0, 20, -24, 0, 0, 42, -50, 0, 0, 80, -92, 0, 0, 147, -172, 0, 0, 260, -296, 0, 0, 445, -510, 0, 0, 744, -840, 0, 0, 1215, -1372, 0, 0, 1944, -2176, 0, 0, 3059, -3424, 0, 0, 4740, -5268, 0, 0, 7239, -8040, 0, 0, 10920

Offset

0,2

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 = 0..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

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

Euler transform of period 4 sequence [ -2, -1, -2, 1, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (72 t)) = (4/3)^(1/2) (t/i)^(-1/2) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A137829.

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

a(4*n + 2) = a(4*n + 3) = 0.

a(n) = (-1)^n * A137828(n). a(4*n) = A051136(n). a(4*n + 1) = -2 * A137829(n).

Example

G.f. = 1 - 2*x + 4*x^4 - 4*x^5 + 9*x^8 - 12*x^9 + 20*x^12 - 24*x^13 + 42*x^16 + ...
G.f. = 1/q - 2*q^2 + 4*q^11 - 4*q^14 + 9*q^23 - 12*q^26 + 20*q^35 - 24*q^38 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] / QPochhammer[ x^4]^2, {x, 0, n}]; (* Michael Somos, Oct 04 2015 *)

Prog

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

Crossrefs

Cf. A051136, A137828, A137829.

Sequence in context: A045836 A182056 A072070 * A137828 A264655 A264476

Adjacent sequences: A137827 A137828 A137829 * A137831 A137832 A137833

Keyword

sign

Author

Michael Somos, Feb 12 2008

Status

approved