A260314 Expansion of phi(x)^2 / phi(-x^2) in powers of x where phi() is a Ramanujan theta function.

1, 4, 6, 8, 16, 24, 32, 48, 66, 92, 128, 168, 224, 296, 384, 496, 640, 816, 1030, 1304, 1632, 2032, 2528, 3120, 3840, 4716, 5760, 7008, 8512, 10296, 12416, 14944, 17922, 21440, 25600, 30480, 36208, 42936, 50784, 59952, 70656, 83088, 97536, 114312, 133728

Offset

0,2

Comments

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

G.f. is a period 1 Fourier series which satisfies f(-1/ (32 t)) = 8^(1/2) (t/i)^(1/2) g(t) where q = exp(2*Pi i t) and g() is the g.f. for A260313.

Links

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

Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of f(-x^4) * chi(x)^4 = psi(x) * chi(x)^3 = phi(-x^2)^3 / phi(-x)^2 = psi(x)^4 / f(-x^4)^3 in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions.

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

Euler transform of period 4 sequence [4, -4, 4, -1, ... ].

a(n) ~ exp(Pi*sqrt(n/2)) / (2*sqrt(2*n)). - Vaclav Kotesovec, Oct 14 2015

Example

G.f. = 1 + 4*x + 6*x^2 + 8*x^3 + 16*x^4 + 24*x^5 + 32*x^6 + 48*x^7 + ...

Mathematica

a[ n_]:= SeriesCoefficient[ EllipticTheta[ 3, 0, x]^2 / EllipticTheta[ 4, 0, x^2], {x, 0, n}];
nmax=60; CoefficientList[Series[Product[(1-x^(2*k))^8 / ((1-x^k)^4 * (1-x^(4*k))^3), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 14 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^8 / (eta(x + A)^4 * eta(x^4 + A)^3), n))};
(PARI) q='q+O('q^99); Vec(eta(q^2)^8/(eta(q)^4*eta(q^4)^3)) \\ Altug Alkan, Mar 19 2018

Crossrefs

Cf. A260313.

Sequence in context: A295006 A269833 A049421 * A238269 A039624 A083166

Adjacent sequences: A260311 A260312 A260313 * A260315 A260316 A260317

Keyword

nonn

Author

Michael Somos, Jul 22 2015

Status

approved