A226864 Expansion of phi(-x^3) * f(-x^4) in powers of x where phi(), f() are Ramanujan theta functions.

1, 0, 0, -2, -1, 0, 0, 2, -1, 0, 0, 2, 2, 0, 0, 0, -2, 0, 0, 0, -1, 0, 0, -2, 0, 0, 0, -2, 1, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, -2, 1, 0, 0, 2, -2, 0, 0, -2, -2, 0, 0, 0, -3, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, -2, 0, 0, 0, 2, 0, 0, 2, 0, 0

Offset

0,4

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

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

G.f.: (Sum_{k in Z} (-1)^k * x^(3*k^2)) * Product_{k>0} (1 - x^(4*k)).

a(n) = (-1)^n * A226862(n). a(4*n + 1) = a(4*n + 2) = 0. a(4*n) = A226289(n).

Example

1 - 2*x^3 - x^4 + 2*x^7 - x^8 + 2*x^11 + 2*x^12 - 2*x^16 - x^20 - 2*x^23 + ...
q - 2*q^19 - q^25 + 2*q^43 - q^49 + 2*q^67 + 2*q^73 - 2*q^97 - q^121 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q^3] QPochhammer[ q^4], {q, 0, n}]

Prog

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

Crossrefs

Cf. A226289, A226862.

Sequence in context: A302236 A262929 A226862 * A257399 A168313 A072575

Adjacent sequences: A226861 A226862 A226863 * A226865 A226866 A226867

Keyword

sign

Author

Michael Somos, Jun 20 2013

Status

approved