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%I #10 Sep 08 2022 08:45:54
%S 1,-6,0,-6,6,0,0,-12,0,-6,0,0,6,-12,0,0,6,0,0,-12,0,-12,0,0,0,-6,0,-6,
%T 12,0,0,-12,0,0,0,0,6,-12,0,-12,0,0,0,-12,0,0,0,0,6,-18,0,0,12,0,0,0,
%U 0,-12,0,0,0,-12,0,-12,6,0,0,-12,0,0,0,0,0,-12,0
%N Expansion of a(-q) in powers of q where a(q) is a cubic AGM function.
%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H Antti Karttunen, <a href="/A180318/b180318.txt">Table of n, a(n) for n = 0..16384</a>
%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 2 * a(q^4) - a(q) in powers of q where a() is a cubic AGM theta function.
%F Expansion of phi(-q) * phi(-q^3) - 4 * q * psi(q^2) * psi(q^6) in powers of q where phi(), psi() are Ramanujan theta functions. - _Michael Somos_, Sep 14 2015
%F Expansion of theta_3(-q) * theta_3(-q^3) - theta_2(q) * theta_2(q^3) in powers of q.
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = - (12)^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
%F a(n) = (-1)^n * A004016(n).
%F G.f.: 1 + 6 * Sum_{k>0} (-x)^k/(1 + (-x)^k + x^(2*k)) = Sum_{j, k in Z} (-x)^(j*j + j*k + k*k).
%F a(2*n) = -6 * A033762(n). a(4*n) = A004016(n). a(4*n + 1) = -6 * A112604(n). a(4*n + 2) = 0. a(4*n + 3) = -6 * A112605(n). - _Michael Somos_, Sep 14 2015
%e G.f. = 1 - 6*q - 6*q^3 + 6*q^4 - 12*q^7 - 6*q^9 + 6*q^12 - 12*q^13 + 6*q^16 + ...
%t a[ n_] := If[ n < 1, Boole[n == 0], (-1)^n 6 Sum[ KroneckerSymbol[ -3, d], {d, Divisors[ n]}]]; (* _Michael Somos_, Sep 14 2015 *)
%t a[ n_] := SeriesCoefficient[ (QPochhammer[ -q]^3 - 9 q QPochhammer[ -q^9]^3) / QPochhammer[ -q^3], {q, 0, n}]; (* _Michael Somos_, Sep 14 2015 *)
%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] EllipticTheta[ 4, 0, q^3] - EllipticTheta[ 2, 0, q] EllipticTheta[ 2, 0, q^3], {q, 0, n}]; (* _Michael Somos_, Sep 14 2015 *)
%o (PARI) {a(n) = if( n<1, n==0, 6 * (-1)^n * sumdiv(n, d, kronecker(d, 3)))};
%o (Magma) A := Basis( ModularForms( Gamma1(12), 1), 75); A[1] - 6*A[2] - 6*A[4] + 6*A[5]; /* _Michael Somos_, Sep 14 2015 */
%Y Cf. A004016, A033762, A112604. A112605.
%K sign
%O 0,2
%A _Michael Somos_, Aug 27 2010
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