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

0,2

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

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

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

Example

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

Mathematica

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

Prog

(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))}

Crossrefs

Cf. A014969, A215348, A232772.

Sequence in context: A295949 A077666 A232358 * A346461 A232392 A176143

Adjacent sequences: A212315 A212316 A212317 * A212319 A212320 A212321

Keyword

nonn

Author

Michael Somos, Oct 25 2013

Status

approved