A014705 Expansion of ((theta_2)^4 + (theta_3)^4) / eta(z/2)^4.

1, 28, 134, 568, 1809, 5316, 13990, 34696, 80724, 180068, 384940, 796760, 1598789, 3127360, 5971922, 11170160, 20491033, 36947444, 65553412, 114619248, 197681341, 336670120, 566630192, 943234040, 1553941445, 2535325644, 4098671374, 6568931200, 10441889389

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

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

Examples

			G.f. = 1 + 28*x + 134*x^2 + 568*x^3 + 1809*x^4 + 5316*x^5 + 13990*x^6 + ...
G.f. = 1/q + 28*q^5 + 134*q^11 + 568*q^17 + 1809*q^23 + 5316*q^29 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A101127.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ (QPochhammer[ x, x^2]^8 + QPochhammer[ -x, x^2]^8 ) / 2, {x, 0, 2 n}]; (* Michael Somos, Sep 30 2013 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q]^4 + EllipticTheta[ 2, 0, q]^4) / QPochhammer[ q]^4, {q, 0, n}]; (* Michael Somos, Sep 30 2013 *)

PARI

{a(n) = if( n<0, 0, polcoeff( prod(k=1, n, 1 + x^(2*k-1), 1 + x * O(x^(2*n)))^8, 2*n))}; /* Michael Somos, Sep 30 2013 */

Formula

Expansion of (phi(x)^4 + 16 * x* psi(x^2)^4) / f(-x)^4 in powers of x where phi(), psi(), f() are Ramanujan theta functions. - Michael Somos, Sep 30 2013

Expansion of (phi(x)^4 + phi(-x)^4) / (2 * f(-x^2)^4) = (chi(x)^8 + chi(-x)^8) / 2 in powers of x^2 where phi(), chi(), f() are Ramanujan theta functions. - Michael Somos, Sep 30 2013

a(n) = A101127(2*n). - Michael Somos, Sep 30 2013

a(n) ~ exp(2*Pi*sqrt(2*n/3)) / (2^(7/4) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Apr 19 2018