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A089807 Expansion of Jacobi theta function (3theta_3(q^9)-theta_3(q))/2.

%I A089807 #52 Jun 28 2025 19:44:54
%S A089807 1,-1,0,0,-1,0,0,0,0,2,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,
%T A089807 0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,
%U A089807 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1
%N A089807 Expansion of Jacobi theta function (3theta_3(q^9)-theta_3(q))/2.
%C A089807 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A089807 This is an example of the quintuple product identity in the form f(a*b^4, a^2/b) - (a/b) * f(a^4*b, b^2/a) = f(-a*b, -a^2*b^2) * f(-a/b, -b^2) / f(a, b) where a = x^5, b = x. - _Michael Somos_, Jul 12 2012
%C A089807 Number 11 of the 14 primitive eta-products which are holomorphic modular forms of weight 1/2 listed by D. Zagier on page 30 of "The 1-2-3 of Modular Forms". - _Michael Somos_, May 04 2016
%H A089807 Seiichi Manyama, <a href="/A089807/b089807.txt">Table of n, a(n) for n = 0..10000</a>
%H A089807 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.
%H A089807 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>.
%H A089807 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/QuintupleProductIdentity.html">Quintuple Product Identity</a>.
%H A089807 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%H A089807 I. J. Zucker, <a href="http://dx.doi.org/10.1088/0305-4470/23/2/009">Further Relations Amongst Infinite Series and Products. II. The Evaluation of Three-Dimensional Lattice Sums</a>, J. Phys. A: Math. Gen. 23, 117-132, 1990.
%F A089807 a(n) = -b(n) where b() is multiplicative with b(3^e) = -2(1 + (-1)^e) / 2 if e>0, b(p^e) = (1 + (-1)^e) / 2 otherwise.
%F A089807 From _Michael Somos_, Nov 05 2005: (Start)
%F A089807 Expansion of eta(q) * eta(q^4) * eta(q^6)^2 / (eta(q^2) * eta(q^3) * eta(q^12)) in powers of q.
%F A089807 Euler transform of period 12 sequence [ -1, 0, 0, -1, -1, -1, -1, -1, 0, 0, -1, -1, ...].
%F A089807 G.f.: (Sum_{k in Z} 3 * x^((3*k)^2) - x^(k^2)) / 2 = Product_{k>0} (1 - x^k) / ((1 - x^(12*k - 2)) * (1 - x^(12*k - 3)) * (1 - x^(12*k - 9)) * (1 - x^(12*k - 10))). (End)
%F A089807 Expansion of Jacobi theta function theta_3(Pi/3, q) in powers of q. - _Michael Somos_, Jan 26 2006
%F A089807 Expansion of chi(q^3) * psi(-q) in powers of q where chi(), psi() are Ramanujan theta functions. - _Michael Somos_, May 19 2007
%F A089807 Expansion of f(x*w, x/w) in powers of x where w is a primitive cube root of unity and f(, ) is Ramanujan's general theta function. - _Michael Somos_, Sep 17 2007
%F A089807 G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 18^(1/2) (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A089801.
%F A089807 a(n) = (-1)^n * A089810(n). - _Michael Somos_, Jan 20 2012
%F A089807 For n > 0, a(n) = (floor(sqrt(n))-floor(sqrt(n-1)))*(2-4*sin(floor(sqrt(n))*Pi/3)^2). - _Mikael Aaltonen_, Jan 17 2015
%F A089807 Sum_{k=1..n} abs(a(k)) ~ (4/3)*sqrt(n). - _Amiram Eldar_, Jan 27 2024
%e A089807 G.f. = 1 - q - q^4 + 2*q^9 - q^16 - q^25 + 2*q^36 - q^49 - q^64 + 2*q^81 + ...
%t A089807 a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, Pi/3, q], {q, 0, n}]; (* _Michael Somos_, Jul 12 2012 *)
%t A089807 a[ n_] := SeriesCoefficient[ (3 EllipticTheta[ 3, 0, q^9] - EllipticTheta[ 3, 0, q])/2, {q, 0, n}]; (* _Michael Somos_, Jul 12 2012 *)
%t A089807 a[ n_] := SeriesCoefficient[ QPochhammer[ -q^3, q^6] EllipticTheta[ 2, 0, Sqrt[ -q]] / (2 (-q)^(1/8)), {q, 0, n}] (* _Michael Somos_, Jul 12 2012 *);
%o A089807 (PARI) {a(n) = if( n<1, n==0, issquare(n) * (3*(n%3==0) - 1))}; /* _Michael Somos_, Nov 05 2005 */
%Y A089807 Related to the 14 primitive eta-products which are holomorphic modular forms of weight 1/2: A000122, A002448, A010054, A010815, A080995, A089801, A089802, this sequence, A089810, A089812, A106459, A121373, A133985, A133988. - _Seiichi Manyama_, May 15 2017
%Y A089807 Cf. A000700, A000122, A010054, A121373.
%K A089807 sign
%O A089807 0,10
%A A089807 _Eric W. Weisstein_, Nov 12 2003