A187149 Expansion of psi(-x)^4 * chi(-x^2)^2 in powers of x where psi(), chi() are Ramanujan theta functions.

1, -4, 4, 0, 2, 0, -8, 0, -5, 16, 4, 0, -10, 0, -8, 0, 9, -8, 0, 0, 14, 0, 16, 0, -10, -32, 4, 0, 0, 0, 8, 0, 14, 20, -20, 0, 2, 0, 0, 0, -11, 16, -20, 0, -32, 0, 16, 0, 0, 40, 4, 0, 14, 0, -8, 0, -9, -32, -20, 0, 26, 0, 0, 0, 2, -36, 28, 0, 0, 0, 16, 0, 16, 0, 28, 0, -22, 0, 0, 0, 14, -56, -16

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(x^4) / eta(x^2))^2 in powers of q.

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

a(n) = (-1)^n * A106508(n). a(4*n + 3) = a(8*n + 5) = 0.

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

Example

G.f. = 1 - 4*x + 4*x^2 + 2*x^4 - 8*x^6 - 5*x^8 + 16*x^9 + 4*x^10 - 10*x^12 + ...
G.f. = q - 4*q^4 + 4*q^7 + 2*q^13 - 8*q^19 - 5*q^25 + 16*q^28 + 4*q^31 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (QPochhammer[ x]^2 QPochhammer[ x^4] / QPochhammer[ x^2])^2, {x, 0, n}]; (* Michael Somos, Sep 02 2015 *)
a[ n_] := SeriesCoefficient[ (1/4) x^(-1/2) EllipticTheta[ 2, Pi/4, x^(1/2)]^4  QPochhammer[ x^2, x^4]^2, {x, 0, n}]; (* Michael Somos, Sep 02 2015 *)

Prog

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

Crossrefs

Cf. A106508.

Sequence in context: A318143 A131124 A131125 * A106508 A177036 A158100

Adjacent sequences: A187146 A187147 A187148 * A187150 A187151 A187152

Keyword

sign

Author

Michael Somos, Mar 05 2011

Status

approved