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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
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
Dalen Dockery, Marie Jameson, James A. Sellers, and Samuel Wilson, d-Fold Partition Diamonds, arXiv:2307.02579 [math.NT], 2023. See p. 14.
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.
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
KEYWORD
sign
AUTHOR
Eric W. Weisstein, Nov 12 2003
STATUS
approved