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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-Fran├žois 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.)