A225564 Expansion of psi(-x)^2 * f(-x^4)^6 in powers of x where psi(), f() are Ramanujan theta functions.

1, -2, 1, -2, -4, 12, -3, 10, -3, -20, -7, -8, 29, -10, 25, -28, -12, 54, 20, 34, -74, -42, -80, 22, 53, 40, -43, 16, 73, -50, 114, -38, -20, -68, 104, -100, -47, 114, -47, -24, -100, -68, -151, 50, 137, 244, -40, 326, -23, -194, -30, 50, -100, -160, 6, -274

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^(-5/4) (eta(q) * eta(q^4)^4 / eta(q^2))^2 in powers of q.

Expansion of chi(-x)^2 * f(-x^4)^8 = psi(-x)^8 / chi(-x)^6 in powers of x where psi(), chi(), f() are Ramanujan theta functions.

Euler transform of period 4 sequence [ -2, 0, -2, -8, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 1024 (t / i)^6 g(t) where q = exp(2 Pi i t) and g() is the g.f. of A215600.

G.f.: Product_{k>0} (1 - x^(4*k))^8 * (1 - x^(2*k - 1))^2.

Example

1 - 2*x + x^2 - 2*x^3 - 4*x^4 + 12*x^5 - 3*x^6 + 10*x^7 - 3*x^8 - 20*x^9 + ...
q^5 - 2*q^9 + q^13 - 2*q^17 - 4*q^21 + 12*q^25 - 3*q^29 + 10*q^33 - 3*q^37 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ q^4]^8 QPochhammer[ q, q^2]^2, {q, 0, n}]
a[ n_] := SeriesCoefficient[ (1/ 16) EllipticTheta[ 2, Pi/4, q^(1/2)]^8 / QPochhammer[ q, q^2]^6, {q, 0, n + 1}]
a[ n_] := SeriesCoefficient[ (1/2) QPochhammer[ q^4]^6 EllipticTheta[ 2, Pi/4, q^(1/2)]^2, {q, 0, n + 1/4}]

Prog

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

Crossrefs

Cf. A215600.

Sequence in context: A000087 A145667 A095067 * A032259 A109851 A371136

Adjacent sequences: A225561 A225562 A225563 * A225565 A225566 A225567

Keyword

sign

Author

Michael Somos, May 17 2013

Status

approved