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A089812
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Expansion of Jacobi theta function q^(-1/8) * (theta_2(q^(1/2)) - 3 * theta_2(q^(9/2))) / 2 in powers of q.
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7
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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
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OFFSET
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0,2
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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^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
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LINKS
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Dalen Dockery, Marie Jameson, James A. Sellers, and Samuel Wilson, d-Fold Partition Diamonds, arXiv:2307.02579 [math.NT], 2023. See p. 14.
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FORMULA
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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
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
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EXAMPLE
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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 + ...
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MATHEMATICA
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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 *)
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PROG
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(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 */
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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