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A089802
Expansion of q^(-1/3) * (theta_4(q^3) - theta_4(q^(1/3))) / 2 in powers of q.
10
1, -1, 0, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
OFFSET
0,1
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Number 10 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
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Eric Weisstein's World of Mathematics, Jacobi Theta Functions
FORMULA
Expansion of q^(-1/3) * (eta(q) * eta(q^6)^2) / (eta(q^2) * eta(q^3)) in powers of q. - Michael Somos, Apr 12 2005
Expansion of chi(-x) * psi(x^3) in powers of x where psi(), chi() are Ramanujan theta functions. - Michael Somos, Dec 23 2011
Expansion of f(-x, -x^5) in powers of x, where f(, ) is Ramanujan's general theta function.
a(n) = b(3*n + 1) where b() is multiplicative with b(3^e) = 0^e, b(2^e) = - (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)) = 8^(1/2) (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A089812. - Michael Somos, Dec 23 2011
Euler transform of period 6 sequence [-1, 0, 0, 0, -1, -1, ...]. - Michael Somos, Apr 12 2005
abs(a(n)) is the characteristic function of A001082. - Michael Somos, Oct 31 2005
G.f.: Sum_{k in Z} (-1)^k * x^((3*k^2 - 2*k)) = Product_{k>0} (1 - x^(6*k)) * (1 - x^(6*k - 1)) * (1 - x^(6*k - 5)). - Michael Somos, Oct 31 2005
A002448(3*n + 1) = -2 * a(n). - Michael Somos, Jul 07 2006
a(n) = (-1)^n * A089801(n).
a(n) = -(1/n)*Sum_{k=1..n} A284362(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017
EXAMPLE
G.f. = 1 - x - x^5 + x^8 + x^16 - x^21 - x^33 + x^40 + x^56 - x^65 - x^85 + ...
G.f. = q - q^4 - q^16 + q^25 + q^49 - q^64 - q^100 + q^121 + q^169 - q^196 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, x^3] - EllipticTheta[ 4, 0, x^(1/3)]) / (2 x^(1/3)), {x, 0, n}]; (* Michael Somos, Jun 30 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2] EllipticTheta[ 2, 0, x^(3/2)] / (2 x^(3/8)), {x, 0, n}]; (* Michael Somos, Jun 30 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2] QPochhammer[ -x^3, x^3] QPochhammer[ x^6], {x, 0, n}]; (* Michael Somos, Jun 30 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^6] QPochhammer[ x^5, x^6] QPochhammer[ x^6], {x, 0, n}]; (* Michael Somos, Jun 30 2015 *)
a[ n_] := (-1)^n Sign @ SquaresR[ 1, 3 n + 1]; (* Michael Somos, Jun 30 2015 *)
PROG
(PARI) {a(n) = (-1)^n * issquare(3*n + 1)}; /* Michael Somos, Apr 12 2005 */
CROSSREFS
KEYWORD
sign
AUTHOR
Eric W. Weisstein, Nov 12 2003
EXTENSIONS
Corrected by N. J. A. Sloane, Nov 05 2005
STATUS
approved