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Expansion of phi(-q)^3 in powers of q where phi() is a Ramanujan theta function.
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%I #24 Sep 08 2022 08:46:02

%S 1,-6,12,-8,6,-24,24,0,12,-30,24,-24,8,-24,48,0,6,-48,36,-24,24,-48,

%T 24,0,24,-30,72,-32,0,-72,48,0,12,-48,48,-48,30,-24,72,0,24,-96,48,

%U -24,24,-72,48,0,8,-54,84,-48,24,-72,96,0,48,-48,24,-72,0,-72,96

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

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

%H Seiichi Manyama, <a href="/A213384/b213384.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..2500 from G. C. Greubel)

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

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

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

%F 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.

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

%F 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.

%e 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 + ...

%t a[ n_] := (-1)^n SquaresR[ 3, n]; (* _Michael Somos_, May 21 2015 *)

%t a[ n_] := (-1)^n Length @ FindInstance[ n == x^2 + y^2 + z^2, {x, y, z}, Integers, 10^9]; (* _Michael Somos_, May 21 2015 *)

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q]^3, {q, 0, n}]; (* _Michael Somos_, May 21 2015 *)

%t a[ n_] := SeriesCoefficient[ (QPochhammer[ q]^2 / QPochhammer[ q^2])^3, {q, 0, n}]; (* _Michael Somos_, May 21 2015 *)

%o (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))};

%o (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 */

%o (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 */

%o (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 */

%o (Julia) # JacobiTheta4 is defined in A002448.

%o A213384List(len) = JacobiTheta4(len, 3)

%o A213384List(63) |> println # _Peter Luschny_, Mar 12 2018

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

%K sign

%O 0,2

%A _Michael Somos_, Jun 10 2012