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A131946 Expansion of (phi(-q) * phi(-q^3))^2 in powers of q where phi() is a Ramanujan theta function. 4

%I

%S 1,-4,4,-4,20,-24,4,-32,52,-4,24,-48,20,-56,32,-24,116,-72,4,-80,120,

%T -32,48,-96,52,-124,56,-4,160,-120,24,-128,244,-48,72,-192,20,-152,80,

%U -56,312,-168,32,-176,240,-24,96,-192,116,-228,124,-72,280,-216,4

%N Expansion of (phi(-q) * phi(-q^3))^2 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).

%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

%D N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 85, Eq. (32.66).

%H G. C. Greubel, <a href="/A131946/b131946.txt">Table of n, a(n) for n = 0..10000</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 (4*a(q^2)^2 - a(q)^2) / 3 in powers of q where a() is a cubic AGM theta function.

%F Expansion of (b(q)^2 / b(q^2)) * (c(q)^2 / c(q^2)) / 3 in powers of q where b(), c() are cubic AGM theta functions.

%F Expansion of (eta(q) * eta(q^3))^4 / ( eta(q^2) * eta(q^6))^2 in powers of q.

%F Euler transform of period 6 sequence [-4, -2, -8, -2, -4, -4, ...].

%F a(n) = -4 * b(n) where b() is multiplicative with b(2^e) = 3 - 2^(e+1), b(3^e) = 1, b(p^e) = (p^(e+1) - 1) / (p - 1) if p>3. - _Michael Somos_, Sep 19 2013

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 48 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A111932. - _Michael Somos_, Sep 19 2013

%F G.f.: 1 - 4 * (Sum_{k>0} k * (-x)^k / (1 - x^k) * Kronecker(9, k)) = (theta_3(-x) * theta_3(-x^3))^2.

%F a(n) = (-1)^n * A034896(n). a(n) = -4 * A131947(n) unless n = 0.

%F a(3*n) = a(n). a(2*n) = A125514(n). - _Michael Somos_, Sep 19 2013

%e G.f. = 1 - 4*q + 4*q^2 - 4*q^3 + 20*q^4 - 24*q^5 + 4*q^6 - 32*q^7 + 52*q^8 + ...

%t a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, q] EllipticTheta[ 4, 0, q^3])^2, {q, 0, n}]; (* _Michael Somos_, Sep 19 2013 *)

%t a[ n_] := SeriesCoefficient[ (QPochhammer[ q] QPochhammer[ q^3])^4 / (QPochhammer[ q^2] QPochhammer[ q^6])^2, {q, 0, n}]; (* _Michael Somos_, Sep 19 2013 *)

%t a[ n_] := If[ n < 1, Boole[n == 0], -4 Sum[ d {0, 1, -1, 0, -1, 1}[[ Mod[ d, 6] + 1]], {d, Divisors @ n}]]; (* _Michael Somos_, Sep 19 2013 *)

%t a[ n_] := If[ n < 1, Boole[n == 0], -4 Sum[ n/d {6, 1, -3, -2, -3, 1}[[ Mod[ d, 6] + 1]], {d, Divisors @ n}]]; (* _Michael Somos_, Sep 19 2013 *)

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^3 + A))^4 / (eta(x^2 + A) * eta(x^6 + A))^2, n))};

%o (PARI) {a(n) = if( n<1, n==0, -4 * sumdiv( n, d, d * [0, 1, -1, 0, -1, 1][d%6 + 1]))}; /* _Michael Somos_, Sep 19 2013 */

%o (Sage) A = ModularForms( Gamma0(6), 2, prec=55) . basis(); A[0] - 4*A[1] + 4*A[2]; # _Michael Somos_, Sep 19 2013

%o (MAGMA) A := Basis( ModularForms( Gamma0(6), 2), 55); A[1] - 4*A[2] + 4*A[3]; /* _Michael Somos_, Nov 11 2015 */

%Y Cf. A034896, A111932, A125514, A131947.

%K sign

%O 0,2

%A _Michael Somos_, Jul 30 2007

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Last modified July 9 16:50 EDT 2020. Contains 335545 sequences. (Running on oeis4.)