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A089812 Expansion of Jacobi theta function q^(-1/8) * (theta_2(q^(1/2)) - 3 * theta_2(q^(9/2))) / 2 in powers of q.

%I A089812 #48 Feb 16 2025 08:32:51
%S A089812 1,-2,0,1,0,0,1,0,0,0,-2,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,-2,0,0,0,0,
%T A089812 0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-2,0,0,0,0,0,0,0,0,0,0,1,
%U A089812 0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-2,0,0,0,0,0,0,0,0,0,0,0,0,0
%N A089812 Expansion of Jacobi theta function q^(-1/8) * (theta_2(q^(1/2)) - 3 * theta_2(q^(9/2))) / 2 in powers of q.
%C A089812 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A089812 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^2, b = x. - _Michael Somos_, Jan 21 2012
%C A089812 Number 7 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_, Jan 01 2015
%H A089812 Seiichi Manyama, <a href="/A089812/b089812.txt">Table of n, a(n) for n = 0..10000</a>
%H A089812 Dalen Dockery, Marie Jameson, James A. Sellers, and Samuel Wilson, <a href="https://arxiv.org/abs/2307.02579">d-Fold Partition Diamonds</a>, arXiv:2307.02579 [math.NT], 2023. See p. 14.
%H A089812 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A089812 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H A089812 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>
%H A089812 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/QuintupleProductIdentity.html">Quintuple Product Identity</a>
%H A089812 D. Zagier, <a href="http://dx.doi.org/10.1007/978-3-540-74119-0">Elliptic modular forms and their applications</a> in "The 1-2-3 of modular forms", Springer-Verlag, 2008.
%H A089812 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 A089812 Expansion of q^(-1/8) * eta(q)^2 * eta(q^6) / ( eta(q^2) * eta(q^3) ) in powers of q. - _Michael Somos_, Nov 05 2005
%F A089812 Expansion of Jacobi theta function q^(-1/4) * theta_1(Pi/6, q) in powers of q^2. - _Michael Somos_, Sep 17 2007
%F A089812 Expansion of f(-x, -x^5) * f(-x) / f(-x^6) = f(x^3, x^6) - x * f(1, x^9) in powers of x where f(, ) is Ramanujan's general theta function.
%F A089812 Expansion of phi(-x^9) / chi(-x^3) - 2 * x * psi(x^9) in powers of x where phi(), chi() are Ramanujan theta functions.
%F A089812 Expansion of phi(-x) / chi(-x^3) in powers of x where phi(), chi() are Ramanujan theta functions. - _Michael Somos_, May 04 2016
%F A089812 Euler transform of period 6 sequence [ -2, -1, -1, -1, -2, -1, ...]. - _Michael Somos_, Nov 05 2005
%F A089812 a(n) = b(8*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(3^e) = -2 * (1 + (-1)^e) / 2 if e>0, b(p^e) = (1 + (-1)^e) / 2 if p>3.
%F A089812 G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 18^(1/2) (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A089802.
%F A089812 G.f.: Sum_{k>0} x^((k^2 - k)/2) - 3 * x^(9(k^2 - k)/2 + 1) = Product_{k>0} (1 - x^k) * (1 - x^(6*k - 1)) * (1 - x^(6*k - 5)). - _Michael Somos_, Nov 05 2005
%F A089812 G.f.: Sum_{k in Z} x^(3*k * (3*k + 1)/2) * ( x^(-3*k) - x^(3*k + 1) ). - _Michael Somos_, Jan 21 2012
%F A089812 A133988(n) = (-1)^n * a(n). Convolution inverse of A101230.
%F A089812 a(n) = (floor(sqrt(2*(n+1))+1/2)-floor(sqrt(2*n)+1/2))*(-2+4*sin((floor(sqrt(2*(n+1))+1/2)+1)*Pi/3)^2). - _Mikael Aaltonen_, Jan 17 2015
%e A089812 G.f. = 1 - 2*x + x^3 + x^6 - 2*x^10 + x^15 + x^21 - 2*x^28 + x^36 + x^45 + ...
%e A089812 G.f. = q - 2*q^9 + q^25 + q^49 - 2*q^81 + q^121 + q^169 - 2*q^225 + q^289 + ...
%t A089812 a[n_] := Boole[ IntegerQ[ Sqrt[8*n + 1]]]*(1 - 3*Boole[ Mod[n, 3] > 0]); Table[a[n], {n, 0, 104}] (* _Jean-François Alcover_, Oct 31 2012, after _Michael Somos_ *)
%t A089812 a[ n_] := SeriesCoefficient[ EllipticTheta[ 1, Pi/6, x^(1/2)] / x^(1/8), {x, 0, n}]; (* _Michael Somos_, Jan 01 2015 *)
%t A089812 a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, Pi/3, x^(1/2)] / x^(1/8), {x, 0, n}]; (* _Michael Somos_, Jan 01 2015 *)
%t A089812 a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, x^(1/2)] - 3 EllipticTheta[ 2, 0, x^(9/2)]) / (2 x^(1/8)), {x, 0, n}]; (* _Michael Somos_, Jan 01 2015 *)
%o A089812 (PARI) {a(n) = if( n<1, n==0, issquare( 8*n + 1) * (1 - 3*(n%3>0)))}; /* _Michael Somos_, Nov 05 2005 */
%o A089812 (PARI) {a(n) = (-1)^(n\3 + n) * ((n + 1)%3) * issquare( 8*n + 1)}; /* _Michael Somos_, Dec 23 2011 */
%o A089812 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^6 + A) / eta(x^2 + A) / eta(x^3 + A), n))}; /* _Michael Somos_, Nov 05 2005 */
%o A089812 (Magma) A := Basis( ModularForms( Gamma1(144), 1/2), 841); A[2] - 2*A[7]; /* _Michael Somos_, Jan 01 2015 */
%Y A089812 Cf. A089802, A101230, A133988.
%K A089812 sign
%O A089812 0,2
%A A089812 _Eric W. Weisstein_, Nov 12 2003