A261240 Expansion of f(-x^6, -x^12)^2 / (f(-x, -x) * f(-x^3, -x^15)) in powers of x where f(, ) is Ramanujan's general theta function.

1, 2, 4, 9, 16, 28, 47, 76, 120, 185, 280, 416, 608, 878, 1252, 1765, 2464, 3408, 4676, 6364, 8600, 11545, 15400, 20424, 26938, 35346, 46152, 59981, 77616, 100016, 128369, 164140, 209120, 265510, 335992, 423840, 533035, 668404, 835804, 1042308, 1296448

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

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^6) * psi(x^3) / (phi(-x) * psi(x^9)) in powers of x where phi(), psi(), f() are Ramanujan theta functions.

Expansion of q^(1/2) * eta(q^2) * eta(q^6)^3 * eta(q^9) / (eta(q)^2 * eta(q^3) * eta(q^18)^2) in powers of q.

Euler transform of period 18 sequence [2, 1, 3, 1, 2, -1, 2, 1, 2, 1, 2, -1, 2, 1, 3, 1, 2, 0, ...].

a(n) = A058647(2*n - 1) = A186115(2*n - 1) = A186964(2*n - 1) = A187020(2*n - 1).

a(n) ~ exp(2*Pi*sqrt(2*n)/3) / (2^(7/4) * sqrt(3) * n^(3/4)). - Vaclav Kotesovec, Oct 13 2015

Example

G.f. = 1 + 2*x + 4*x^2 + 9*x^3 + 16*x^4 + 28*x^5 + 47*x^6 + 76*x^7 + ...
G.f. = q^-1 + 2*q + 4*q^3 + 9*q^5 + 16*q^7 + 28*q^9 + 47*q^11 + ...

Mathematica

a[ n_] := SeriesCoefficient[ x^(3/4) QPochhammer[ x^6] EllipticTheta[ 2, 0, x^(3/2)] / (EllipticTheta[ 4, 0, x] EllipticTheta[ 2, 0, x^(9/2)]), {x, 0, n}];
nmax=60; CoefficientList[Series[Product[(1+x^k) * (1-x^(6*k))^3 * (1-x^(9*k)) / ((1-x^k) * (1-x^(3*k)) * (1-x^(18*k))^2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 13 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^6 + A)^3 * eta(x^9 + A) / (eta(x + A)^2 * eta(x^3 + A) * eta(x^18 + A)^2), n))};

Crossrefs

Cf. A058647, A186115, A186964, A187020.

Sequence in context: A114080 A090676 A333779 * A000291 A081055 A034446

Adjacent sequences: A261237 A261238 A261239 * A261241 A261242 A261243

Keyword

nonn

Author

Michael Somos, Aug 12 2015

Status

approved