A279320 Expansion of chi(-x^4) * psi(x^6) / phi(-x) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.

1, 2, 4, 8, 15, 26, 45, 74, 119, 188, 291, 442, 664, 982, 1435, 2076, 2972, 4214, 5929, 8272, 11457, 15762, 21543, 29264, 39532, 53110, 70988, 94430, 125033, 164826, 216388, 282940, 368552, 478326, 618621, 797376, 1024485, 1312184, 1675657, 2133664, 2709307

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 f(-x^2, -x^10) * f(-x^8) / f(-x)^2 in powers of x where f(, ) is Ramanujan's general theta function.

Expansion of q^(-11/12) * eta(q^2) * eta(q^8) * eta(q^12)^2 / (eta(q)^2 * eta(q^4) * eta(q^6)) in powers of q.

Euler transform of period 24 sequence [ 2, 1, 2, 2, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 2, 2, 1, 2, 0, ...].

a(n) ~ exp(sqrt(13*n/3)*Pi/2) * 13^(1/4) / (16*sqrt(2)*3^(3/4)*n^(3/4)). - Vaclav Kotesovec, Dec 10 2016

Example

G.f. = 1 + 2*x + 4*x^2 + 8*x^3 + 15*x^4 + 26*x^5 + 45*x^6 + 74*x^7 + ...
G.f. = q^11 + 2*q^23 + 4*q^35 + 8*q^47 + 15*q^59 + 26*q^71 + 45*q^83 + ...

Mathematica

a[ n_] := SeriesCoefficient[ x^(-3/4)/2 QPochhammer[ -x^4, x^4] EllipticTheta[ 2, 0, x^3] / EllipticTheta[ 4, 0, x], {x, 0, n}];
a[ n_] := SeriesCoefficient[ 2^(-3/2)/x EllipticTheta[ 2, Pi/4, x] EllipticTheta[ 2, 0, x^3] / QPochhammer[ x]^2, {x, 0, n}];
nmax = 50; CoefficientList[Series[Product[(1+x^k) * (1+x^(4*k)) * (1+x^(6*k)) * (1-x^(12*k)) / (1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Dec 10 2016 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^8 + A) * eta(x^12 + A)^2 / (eta(x + A)^2 * eta(x^4 + A) * eta(x^6 + A)), n))};

Crossrefs

Cf. A131945.

Sequence in context: A179001 A222147 A003241 * A182844 A191630 A125513

Adjacent sequences: A279317 A279318 A279319 * A279321 A279322 A279323

Keyword

nonn

Author

Michael Somos, Dec 09 2016

Status

approved