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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. 7
 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, 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, 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 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^2, b = x. - Michael Somos, Jan 21 2012 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 LINKS Seiichi Manyama, Table of n, a(n) for n = 0..10000 Michael Somos, Introduction to Ramanujan theta functions 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 D. Zagier, Elliptic modular forms and their applications in "The 1-2-3 of modular forms", Springer-Verlag, 2008. 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 q^(-1/8) * eta(q)^2 * eta(q^6) / ( eta(q^2) * eta(q^3) ) in powers of q. - Michael Somos, Nov 05 2005 Expansion of Jacobi theta function q^(-1/4) * theta_1(Pi/6, q) in powers of q^2. - Michael Somos, Sep 17 2007 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. Expansion of phi(-x^9) / chi(-x^3) - 2 * x * psi(x^9) in powers of x where phi(), chi() are Ramanujan theta functions. Expansion of phi(-x) / chi(-x^3) in powers of x where phi(), chi() are Ramanujan theta functions. - Michael Somos, May 04 2016 Euler transform of period 6 sequence [ -2, -1, -1, -1, -2, -1, ...]. - Michael Somos, Nov 05 2005 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. 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. 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 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 A133988(n) = (-1)^n * a(n). Convolution inverse of A101230. 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 EXAMPLE G.f. = 1 - 2*x + x^3 + x^6 - 2*x^10 + x^15 + x^21 - 2*x^28 + x^36 + x^45 + ... G.f. = q - 2*q^9 + q^25 + q^49 - 2*q^81 + q^121 + q^169 - 2*q^225 + q^289 + ... MATHEMATICA 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 *) a[ n_] := SeriesCoefficient[ EllipticTheta[ 1, Pi/6, x^(1/2)] / x^(1/8), {x, 0, n}]; (* Michael Somos, Jan 01 2015 *) a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, Pi/3, x^(1/2)] / x^(1/8), {x, 0, n}]; (* Michael Somos, Jan 01 2015 *) 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 *) PROG (PARI) {a(n) = if( n<1, n==0, issquare( 8*n + 1) * (1 - 3*(n%3>0)))}; /* Michael Somos, Nov 05 2005 */ (PARI) {a(n) = (-1)^(n\3 + n) * ((n + 1)%3) * issquare( 8*n + 1)}; /* Michael Somos, Dec 23 2011 */ (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 */ (MAGMA) A := Basis( ModularForms( Gamma1(144), 1/2), 841); A[2] - 2*A[7]; /* Michael Somos, Jan 01 2015 */ CROSSREFS Cf. A089802, A101230, A133988. Sequence in context: A171368 A322353 A133988 * A260942 A260162 A123858 Adjacent sequences:  A089809 A089810 A089811 * A089813 A089814 A089815 KEYWORD sign AUTHOR Eric W. Weisstein, Nov 12 2003 STATUS approved

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Last modified July 26 17:37 EDT 2021. Contains 346294 sequences. (Running on oeis4.)