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 A089807 Expansion of Jacobi theta function (3theta_3(q^9)-theta_3(q))/2. 5
 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, 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, 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,10 COMMENTS Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). 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 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 LINKS Seiichi Manyama, Table of n, a(n) for n = 0..10000 Eric Weisstein's World of Mathematics, Ramanujan Theta Functions Eric Weisstein's World of Mathematics, Jacobi Theta Functions Eric Weisstein's World of Mathematics, Quintuple Product Identity I. J. Zucker, Further Relations Amongst Infinite Series and Products. II. The Evaluation of Three-Dimensional Lattice Sums, J. Phys. A: Math. Gen. 23, 117-132, 1990. FORMULA 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. Expansion of Jacobi theta function theta_3(Pi/3, q) in powers of q. - Michael Somos, Jan 26, 2006 Expansion of chi(q^3) * psi(-q) in powers of q where chi(), psi() are Ramanujan theta functions. - Michael Somos, May 19 2007 Expansion of eta(q) * eta(q^4) * eta(q^6)^2 / (eta(q^2) * eta(q^3) * eta(q^12)) in powers of q. - Michael Somos, Nov 05 2005 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 Euler transform of period 12 sequence [ -1, 0, 0, -1, -1, -1, -1, -1, 0, 0, -1, -1, ...]. - Michael Somos, Nov 05 2005 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. 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))) - Michael Somos, Nov 05 2005 a(n) = (-1)^n * A080910(n). - Michael Somos, Jan 20 2012 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 EXAMPLE G.f. = 1 - q - q^4 + 2*q^9 - q^16 - q^25 + 2*q^36 - q^49 - q^64 + 2*q^81 + ... MATHEMATICA a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, Pi/3, q], {q, 0, n}]; (* Michael Somos, Jul 12 2012 *) a[ n_] := SeriesCoefficient[ (3 EllipticTheta[ 3, 0, q^9] - EllipticTheta[ 3, 0, q])/2, {q, 0, n}]; (* Michael Somos, Jul 12 2012 *) 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 *); PROG (PARI) {a(n) = if( n<1, n==0, issquare(n) * (3*(n%3==0) - 1))}; /* Michael Somos, Nov 05 2005 */ CROSSREFS 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 Sequence in context: A060478 A088806 A280618 * A089810 A214411 A324179 Adjacent sequences:  A089804 A089805 A089806 * A089808 A089809 A089810 KEYWORD sign AUTHOR Eric W. Weisstein, Nov 12 2003 STATUS approved

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Last modified May 28 21:37 EDT 2020. Contains 334690 sequences. (Running on oeis4.)