A208451 Expansion of phi(q) * phi(-q)^3 in powers of q where phi() is a Ramanujan theta function.

1, -4, 0, 16, -8, -24, 0, 32, 24, -52, 0, 48, -32, -56, 0, 96, 24, -72, 0, 80, -48, -128, 0, 96, 96, -124, 0, 160, -64, -120, 0, 128, 24, -192, 0, 192, -104, -152, 0, 224, 144, -168, 0, 176, -96, -312, 0, 192, 96, -228, 0, 288, -112, -216, 0, 288, 192, -320

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 phi(-q^4)^4 - 4 * q * psi(-q^2)^4 = phi(q) * phi(-q)^3 = phi(-q)^2 * phi(-q^2)^2 = phi(-q^2)^6 / phi(q)^2 = psi(-q)^4 * chi(-q^2)^6 = f(-q)^4 * chi(-q^2)^2 = f(-q)^6 / psi(-q)^2 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.

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

Euler transform of period 4 sequence [ -4, -6, -4, -4, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 5128 (t/i)^2 g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A097723.

a(4*n + 2) = 0. a(2*n + 1) = -4 * A121613(n). a(4*n) = A096727(n). a(4*n + 1) = -4 * A112610(n). a(4*n + 3) = 16 * A097723(n). a(8*n) = A004011(n). a(8*n + 4) = -8 * A008438(n).

Example

G.f. = 1 - 4*q + 16*q^3 - 8*q^4 - 24*q^5 + 32*q^7 + 24*q^8 - 52*q^9 + 48*q^11 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (QPochhammer[ q]^2 QPochhammer[ q^2] / QPochhammer[ q^4])^2, {q, 0, n}]; (* Michael Somos, Aug 21 2014 *)

Prog

(PARI) {a(n) = if( n<1, n==0, if( n%4 == 2, 0, -4 * if( n%2, (-1)^(n\2) * sigma(n), -2 * (-1)^(n\4) * sumdiv( n\4, d, if( d%4, d)))))};
(PARI) {a(n) = local(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))};
(Magma) A := Basis( ModularForms( Gamma0(16), 2), 58); A[1] - 4*A[2] + 16*A[4] - 8*A[5]; /* Michael Somos, Aug 21 2014 */

Crossrefs

Cf. A004011, A008438, A096727, A097723, A112610, A121613.

Sequence in context: A167361 A321610 A167314 * A207541 A158802 A230280

Adjacent sequences: A208448 A208449 A208450 * A208452 A208453 A208454

Keyword

sign

Author

Michael Somos, Feb 26 2012

Status

approved