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 A089810 Expansion of Jacobi theta function theta_3(Pi/6, q) in powers of q. 6
 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, 0 (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 8 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 Expansion of Jacobi theta function (3theta_4(q^9) - theta_4(q)) / 2 in powers of q. a(n) is multiplicative with  a(0)=1, a(2^e) = -(1 + (-1)^e)/2, if e>0, a(3^e) = -2(1 + (-1)^e)/2 if e>0, a(p^e) = (1 + (-1)^e)/2 otherwise. Euler transform of period 6 sequence [ 1, -1, 0, -1, 1, -1, ...]. - Michael Somos, Nov 05 2005 G.f.: (Sum_{k in Z} 3 * (-x)^((3*k)^2) - (-x)^(k^2)) / 2 = Product_{k>0} (1 - x^(2*k)) / ((1 - x^(6*k - 1)) * (1 - x^(6*k-5))) - Michael Somos, Nov 05 2005 Expansion of eta(q^2)^2 * eta(q^3) / (eta(q) * eta(q^6)) in powers of q. - Michael Somos, Nov 05 2005 Expansion of psi(q) * chi(-q^3) in powers of q where psi(), chi() are Ramanujan theta functions. - Michael Somos, Sep 16 2007 Expansion of (3 * phi(-q^9) - phi(-q)) / 2 in powers of q where phi() is a Ramanujan theta function. Expansion of Jacobi theta function theta_3(Pi/6, q) in powers of q. - Michael Somos, Sep 17 2007 Expansion of f(x*w, x/w) in powers of x where w is a primitive sixth root of unity and f() is Ramanujan's two-variable theta function. - Michael Somos, Sep 17 2007 G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 72^(1/2) (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A080995. - Michael Somos, Jan 26 2008 G.f.: Product_{k>0} (1 - x^(2*k)) / (1 - x^k + x^(2*k)). - Michael Somos, Jan 26 2008 a(3*n + 2) = a(4*n + 2) = a(4*n + 3) = a(5*n + 2) = a(5*n + 3) = a(8*n + 5) = a(9*n + 3) = a(9*n + 6) = 0. a(3*n + 1) = A089802(n). a(4*n) = A089807(n). a(9*n) = A002448(n). a(n) = (floor(sqrt(n))-floor(sqrt(n-1)))*(abs(2-4*sin((floor(sqrt(n))+1)*Pi/3)^2)-4*sin((floor(sqrt(n))+2)*Pi/3)^2)*(-1)^floor(floor(sqrt(n)-1)/3). Mikael Aaltonen, Jan 17 2015 a(n) = (-1)^n * A089807(n) = A204843(4*n) = A204853(4*n). - Michael Somos, May 25 2015 Convolution square is A258279. - Michael Somos, May 25 2015 a(8*n + 1) = A089812(n). a(12*n + 4) = - A089801(n). - Michael Somos, May 25 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/6, q], {q, 0, n}]; (* Michael Somos, Nov 14 2011 *) a[ n_] := SeriesCoefficient[ (3 EllipticTheta[ 4, 0, q^9] - EllipticTheta[ 4, 0, q])  /2, {q, 0, n}]; (* Michael Somos, Nov 14 2011 *) QP = QPochhammer; s = QP[q^2]^2*(QP[q^3] / (QP[q]*QP[q^6])) + O[q]^100; CoefficientList[s, q] (* Jean-François Alcover, Nov 09 2015, adapted from PARI *) PROG (PARI) {a(n) = my(x); if( n<1, n==0, issquare(n, &x) * (1 + (n%3==0)) * (-1)^((1 + x) \ 3))}; /* Michael Somos, Nov 05 2005 */ (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A) / (eta(x + A) * eta(x^6 + A)), n))}; /* Michael Somos, Jan 26 2008 */ CROSSREFS Cf. A002448, A080995, A089801, A089802, A089807, A089812, A204843, A204853, A258279. Sequence in context: A088806 A280618 A089807 * A214411 A324179 A216577 Adjacent sequences:  A089807 A089808 A089809 * A089811 A089812 A089813 KEYWORD sign,mult AUTHOR Eric W. Weisstein, Nov 12 2003 STATUS approved

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Last modified June 2 10:57 EDT 2020. Contains 334771 sequences. (Running on oeis4.)