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%I #47 Mar 12 2021 22:24:42
%S 1,-1,0,0,0,-1,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,
%T -1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,
%U 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N Expansion of q^(-1/3) * (theta_4(q^3) - theta_4(q^(1/3))) / 2 in powers of q.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C Number 10 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 Seiichi Manyama, <a href="/A089802/b089802.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>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>
%H 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 Expansion of q^(-1/3) * (eta(q) * eta(q^6)^2) / (eta(q^2) * eta(q^3)) in powers of q. - _Michael Somos_, Apr 12 2005
%F Expansion of chi(-x) * psi(x^3) in powers of x where psi(), chi() are Ramanujan theta functions. - _Michael Somos_, Dec 23 2011
%F Expansion of f(-x, -x^5) in powers of x, where f(, ) is Ramanujan's general theta function.
%F a(n) = b(3*n + 1) where b() is multiplicative with b(3^e) = 0^e, b(2^e) = - (1 + (-1)^e) / 2 if e>0, b(p^e) = (1 + (-1)^e) / 2 if p>3.
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 8^(1/2) (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A089812. - _Michael Somos_, Dec 23 2011
%F Euler transform of period 6 sequence [-1, 0, 0, 0, -1, -1, ...]. - _Michael Somos_, Apr 12 2005
%F abs(a(n)) is the characteristic function of A001082. - _Michael Somos_, Oct 31 2005
%F G.f.: Sum_{k in Z} (-1)^k * x^((3*k^2 - 2*k)) = Product_{k>0} (1 - x^(6*k)) * (1 - x^(6*k - 1)) * (1 - x^(6*k - 5)). - _Michael Somos_, Oct 31 2005
%F A002448(3*n + 1) = -2 * a(n). - _Michael Somos_, Jul 07 2006
%F a(n) = (-1)^n * A089801(n).
%F a(n) = -(1/n)*Sum_{k=1..n} A284362(k)*a(n-k), a(0) = 1. - _Seiichi Manyama_, Mar 25 2017
%e G.f. = 1 - x - x^5 + x^8 + x^16 - x^21 - x^33 + x^40 + x^56 - x^65 - x^85 + ...
%e G.f. = q - q^4 - q^16 + q^25 + q^49 - q^64 - q^100 + q^121 + q^169 - q^196 + ...
%t a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, x^3] - EllipticTheta[ 4, 0, x^(1/3)]) / (2 x^(1/3)), {x, 0, n}]; (* _Michael Somos_, Jun 30 2015 *)
%t a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2] EllipticTheta[ 2, 0, x^(3/2)] / (2 x^(3/8)), {x, 0, n}]; (* _Michael Somos_, Jun 30 2015 *)
%t a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2] QPochhammer[ -x^3, x^3] QPochhammer[ x^6], {x, 0, n}]; (* _Michael Somos_, Jun 30 2015 *)
%t a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^6] QPochhammer[ x^5, x^6] QPochhammer[ x^6], {x, 0, n}]; (* _Michael Somos_, Jun 30 2015 *)
%t a[ n_] := (-1)^n Sign @ SquaresR[ 1, 3 n + 1]; (* _Michael Somos_, Jun 30 2015 *)
%o (PARI) {a(n) = (-1)^n * issquare(3*n + 1)}; /* _Michael Somos_, Apr 12 2005 */
%Y Cf. A001082, A002448, A089801, A089812.
%K sign
%O 0,1
%A _Eric W. Weisstein_, Nov 12 2003
%E Corrected by _N. J. A. Sloane_, Nov 05 2005
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