A232772 Expansion of (psi(x)^2 / (phi(-x) * phi(x^2)))^2 in powers of x where phi(), psi() are Ramanujan theta functions.

1, 8, 30, 80, 197, 472, 1046, 2160, 4306, 8360, 15712, 28656, 51127, 89552, 153926, 259904, 432336, 709728, 1150142, 1841200, 2915546, 4570904, 7097622, 10921184, 16664073, 25228176, 37907758, 56553936, 83806768, 123405752, 180611558, 262799248, 380275604

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..2500

Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(-1/2) * (eta(q^2)^7 * eta(q^8)^2 / (eta(q)^4 * eta(q^4)^5))^2 in powers of q.

Euler transform of period 8 sequence [ 8, -6, 8, 4, 8, -6, 8, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 8 g(t) where q = eqp(2 Pi i t) and g() is the g.f. of A233458.

a(n) = A215349(2*n + 1) = A215348(2*n + 1). 2 * a(n) = A212318(2*n + 1) = - A232358(2*n + 1).

a(n) ~ exp(sqrt(2*n)*Pi) / (2^(17/4) * n^(3/4)). - Vaclav Kotesovec, Oct 13 2015

Example

G.f. = 1 + 8*x + 30*x^2 + 80*x^3 + 197*x^4 + 472*x^5 + 1046*x^6 + 2160*x^7 + ...
G.f. = q + 8*q^3 + 30*q^5 + 80*q^7 + 197*q^9 + 472*q^11 + 1046*q^13 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q^(1/2)]^4 / (16 q^(1/2)) / (EllipticTheta[ 4, 0, q] EllipticTheta[ 3, 0, q^2])^2, {q, 0, n}]
nmax=60; CoefficientList[Series[Product[((1-x^k)^3 * (1+x^k)^7 * (1+x^(4*k))^2 / (1-x^(4*k))^3)^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 13 2015 *)

Prog

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^7 * eta(x^8 + A)^2 / (eta(x + A)^4 * eta(x^4 + A)^5))^2, n))}

Crossrefs

Cf. A212318, A215348, A215348, A232358, A233458.

Sequence in context: A348461 A002417 A126858 * A213776 A113751 A343520

Adjacent sequences: A232769 A232770 A232771 * A232773 A232774 A232775

Keyword

nonn

Author

Michael Somos, Nov 30 2013

Status

approved