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A089810
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Expansion of Jacobi theta function theta_3(Pi/6, q) in powers of q.
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6
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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
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OFFSET
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0,10
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COMMENTS
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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
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LINKS
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FORMULA
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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, ...].
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))).
Expansion of eta(q^2)^2 * eta(q^3) / (eta(q) * eta(q^6)) in powers of q. (End)
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.
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. (End)
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.
G.f.: Product_{k>0} (1 - x^(2*k)) / (1 - x^k + x^(2*k)). (End)
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
Sum_{k=1..n} abs(a(k)) ~ (4/3)*sqrt(n). - Amiram Eldar, Jan 27 2024
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EXAMPLE
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G.f. = 1 + q - q^4 - 2*q^9 - q^16 + q^25 + 2*q^36 + q^49 - q^64 - 2*q^81 + ...
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MATHEMATICA
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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 *)
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PROG
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(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 */
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CROSSREFS
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KEYWORD
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sign,mult
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AUTHOR
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STATUS
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approved
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