A213384 Expansion of phi(-q)^3 in powers of q where phi() is a Ramanujan theta function.

1, -6, 12, -8, 6, -24, 24, 0, 12, -30, 24, -24, 8, -24, 48, 0, 6, -48, 36, -24, 24, -48, 24, 0, 24, -30, 72, -32, 0, -72, 48, 0, 12, -48, 48, -48, 30, -24, 72, 0, 24, -96, 48, -24, 24, -72, 48, 0, 8, -54, 84, -48, 24, -72, 96, 0, 48, -48, 24, -72, 0, -72, 96

Offset

0,2

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..2500 from G. C. Greubel)

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

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

Euler transform of period 2 sequence [ -6, -3, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 2^(15/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A008443.

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

a(n) = (-1)^n * A005875(n). a(2*n) = A004015(n). a(2*n + 1) = -2 * A045826(n). a(4*n) = A005875(n). a(4*n + 1) = -6 * A045834(n). a(4*n + 2) = 12 * A045828(n). a(8*n + 3) = -8 * A008443(n). a(8*n + 7) = 0.

Example

G.f. = 1 - 6*q + 12*q^2 - 8*q^3 + 6*q^4 - 24*q^5 + 24*q^6 + 12*q^8 - 30*q^9 + ...

Mathematica

a[ n_] := (-1)^n SquaresR[ 3, n]; (* Michael Somos, May 21 2015 *)
a[ n_] := (-1)^n Length @ FindInstance[ n == x^2 + y^2 + z^2, {x, y, z}, Integers, 10^9]; (* Michael Somos, May 21 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q]^3, {q, 0, n}]; (* Michael Somos, May 21 2015 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ q]^2 / QPochhammer[ q^2])^3, {q, 0, n}]; (* Michael Somos, May 21 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))^3, n))};
(PARI) {a(n) = if( n<0, 0, polcoeff( sum( k=1, sqrtint(n), 2 * (-x)^k^2, 1 + x * O(x^n))^3, n))}; /* Michael Somos, May 21 2015 */
(PARI) {a(n) = my(G); if( n<0, 0, G = [ 1, 0, 0; 0, 1, 0; 0, 0, 1]; (-1)^n * polcoeff( 1 + 2 * x * Ser( qfrep( G, n)), n))}; /* Michael Somos, May 21 2015 */
(Magma) A := Basis( ModularForms( Gamma0(16), 3/2), 63); A[1] - 6*A[2] + 12*A[3] - 8*A[4]; /* Michael Somos, May 21 2015 */
(Julia) # JacobiTheta4 is defined in A002448.
A213384List(len) = JacobiTheta4(len, 3)
A213384List(63) |> println # Peter Luschny, Mar 12 2018

Crossrefs

Cf. A004015, A005875, A008443, A045826, A045834.

Sequence in context: A074590 A272966 A105730 * A005875 A236933 A236927

Adjacent sequences: A213381 A213382 A213383 * A213385 A213386 A213387

Keyword

sign

Author

Michael Somos, Jun 10 2012

Status

approved