A212318 Expansion of phi(q^2)^2 / (phi(-q) * phi(q^4)) in powers of q where phi() is a Ramanujan theta function.

1, 2, 8, 16, 32, 60, 96, 160, 256, 394, 624, 944, 1408, 2092, 3008, 4320, 6144, 8612, 12072, 16720, 22976, 31424, 42528, 57312, 76800, 102254, 135728, 179104, 235264, 307852, 400704, 519808, 671744, 864672, 1109904, 1419456, 1809568, 2300284, 2914272, 3682400

Offset: 0

Author

Michael Somos, Oct 25 2013

Keywords

• nonn

Comments

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

Examples

			G.f. = 1 + 2*q + 8*q^2 + 16*q^3 + 32*q^4 + 60*q^5 + 96*q^6 + 160*q^7 + ...
		

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. A014969, A215348, A232772.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^2]^2 / (EllipticTheta[ 4, 0, q] EllipticTheta[ 3, 0, q^4]), {q, 0, n}]

PARI

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A)^12 * eta(x^16 + A)^2 / (eta(x + A)^2 * eta(x^2 + A)^3 * eta(x^8 + A)^9), n))}

Formula

Expansion of chi(q^2)^5 / (chi(-q) * chi(q^4))^2 in powers of q where chi() is a Ramanujan theta function.

Expansion of eta(q^4)^12 * eta(q^16)^2 / (eta(q)^2 * eta(q^2)^3 * eta(q^8)^9) in powers of q.

Euler transform of period 16 sequence [ 2, 5, 2, -7, 2, 5, 2, 2, 2, 5, 2, -7, 2, 5, 2, 0, ...].

a(n) = 2 * A215348(n) unless n=0. a(2*n) = A014969(n). a(2*n + 1) = 2 * A232772(n).

a(n) ~ exp(sqrt(n)*Pi)/(4*sqrt(2)*n^(3/4)). - Vaclav Kotesovec, Sep 08 2017