A034951 Expansion of eta(8z)*eta(16z)*theta_3(2z)*theta_3(4z).

1, 2, 2, 4, 1, -2, 2, -4, -2, 2, -8, -4, -1, -4, -6, 0, -4, -8, 10, -4, -6, 6, 2, 8, 9, -4, -6, 4, 4, 14, 2, 4, 4, 10, 8, -12, 14, -2, 8, 8, -11, -6, -4, 12, -2, -8, 0, -4, -2, -2, -6, 4, -16, -2, -6, -20, 2, 8, 2, -8, -7, -12, -12, -16, 12, -6, -8, 8, 10, -10, -16, 4, -12, 18, 18, -4, -2, 0, 18, 12, -16, 2, -8, 20, -9, 2, 18, -4, 28, -6, 2

Offset

0,2

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Johann Cigler, Some Pascal-like triangles, 2018.

Ken Ono and Christopher Skinner, Fourier Coefficients of Half-Integral Weight Modular Forms Modulo l, Ann. Math., 147 (1998), 453-470.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

G.f.: Product_{k>0} (1 - x^(2*k)) * (1 + x^k)^2 * (1 - x^(4*k))^3 / (1 + x^(4*k)). - Michael Somos, Sep 21 2005

Euler transform of period 8 sequence [2, -1, 2, -5, 2, -1, 2, -4, ...]. - Michael Somos, Sep 21 2005

Expansion of q^(-1/2) * (eta(q^2)^3 * eta(q^4)^4) / (eta(q)^2 * eta(q^8)) in powers of q. - Michael Somos, Sep 21 2005

Expansion of phi(x) * f(x^2)^2 * f(-x^8) = psi(x)^2 * f(x^2) * f(-x^4) = psi(x)^2 * psi(-x^2) * phi(x^2) = psi(x^2)^2 * phi(x) * phi(-x^4) = psi(x)^2 * psi(x^2) * phi(-x^4) in powers of x where phi(), psi(), f() are Ramanujan theta functions. - Michael Somos, Jul 07 2014

a(31*n + 15) = 0 unless n == 15 (mod 31). a(961*n + 480) = -31 * a(n). - Michael Somos, Jul 07 2014

Example

G.f. = 1 + 2*x + 2*x^2 + 4*x^3 + x^4 - 2*x^5 + 2*x^6 - 4*x^7 - 2*x^8 + ...
G.f. = q + 2*q^3 + 2*q^5 + 4*q^7 + q^9 - 2*q^11 + 2*q^13 - 4*q^15 - 2*q^17 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x]^2 EllipticTheta[ 3, 0, x] EllipticTheta[ 4, 0, x^4] / (4 x^(1/2)), {x, 0, n}];
QP = QPochhammer; s = (QP[q^2]^3*QP[q^4]^4)/(QP[q]^2*QP[q^8]) + O[q]^90; CoefficientList[s, q] (* Jean-François Alcover, Nov 14 2015, adapted from PARI *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^4 + A)^4 / (eta(x + A)^2 * eta(x^8 + A)), n))}; /* Michael Somos, Sep 21 2005 */

Crossrefs

Sequence in context: A099320 A206714 A230442 * A317826 A317836 A214740

Adjacent sequences: A034948 A034949 A034950 * A034952 A034953 A034954

Keyword

sign

Author

N. J. A. Sloane

Status

approved