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

%I A213384 #25 Feb 16 2025 08:33:17
%S A213384 1,-6,12,-8,6,-24,24,0,12,-30,24,-24,8,-24,48,0,6,-48,36,-24,24,-48,
%T A213384 24,0,24,-30,72,-32,0,-72,48,0,12,-48,48,-48,30,-24,72,0,24,-96,48,
%U A213384 -24,24,-72,48,0,8,-54,84,-48,24,-72,96,0,48,-48,24,-72,0,-72,96
%N A213384 Expansion of phi(-q)^3 in powers of q where phi() is a Ramanujan theta function.
%C A213384 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A213384 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 A213384 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A213384 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A213384 Expansion of (eta(q)^2 / eta(q^2))^3 in powers of q.
%F A213384 Euler transform of period 2 sequence [ -6, -3, ...].
%F A213384 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 A213384 G.f.: (Sum_{k in Z} (-1)^k * x^k^2)^3.
%F A213384 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 A213384 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 A213384 a[ n_] := (-1)^n SquaresR[ 3, n]; (* _Michael Somos_, May 21 2015 *)
%t A213384 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 A213384 a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q]^3, {q, 0, n}]; (* _Michael Somos_, May 21 2015 *)
%t A213384 a[ n_] := SeriesCoefficient[ (QPochhammer[ q]^2 / QPochhammer[ q^2])^3, {q, 0, n}]; (* _Michael Somos_, May 21 2015 *)
%o A213384 (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 A213384 (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 A213384 (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 A213384 (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 A213384 (Julia) # JacobiTheta4 is defined in A002448.
%o A213384 A213384List(len) = JacobiTheta4(len, 3)
%o A213384 A213384List(63) |> println # _Peter Luschny_, Mar 12 2018
%Y A213384 Cf. A004015, A005875, A008443, A045826, A045834.
%K A213384 sign
%O A213384 0,2
%A A213384 _Michael Somos_, Jun 10 2012