A128770 Expansion of phi(-q^9) / phi(-q) in powers of q where phi() is a Ramanujan theta function.

1, 2, 4, 8, 14, 24, 40, 64, 100, 152, 228, 336, 488, 700, 992, 1392, 1934, 2664, 3640, 4936, 6648, 8896, 11832, 15648, 20584, 26942, 35096, 45512, 58768, 75576, 96816, 123568, 157156, 199200, 251676, 316992, 398072, 498460, 622448, 775216, 963012

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], 2015-2016.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

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

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

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u-1) * (v^2-u) - 2*u*v * (1-v).

G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (v-u)^3 - v*(3*u-1) * (1-v) * (1 - 2*v + 3*u*v).

G.f.: Product_{k>0} (1 + x^k) * (1 - x^(9*k)) / ( (1 - x^k) * (1+x^(9*k)) ).

Convolution inverse of A128771. a(n) = 2*A128129(n) unles n = 0.

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

Example

G.f. = 1 + 2*x + 4*x^2 + 8*x^3 + 14*x^4 + 24*x^5 + 40*x^6 + 64*x^7 + 100*x^8 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q^9] / EllipticTheta[ 4, 0, q], {q, 0, n}]; (* Michael Somos, Apr 26 2015 *)
nmax=60; CoefficientList[Series[Product[(1+x^k) * (1-x^(9*k)) / ((1-x^k) * (1+x^(9*k))), {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^9 + A)^2 / (eta(x + A)^2 * eta(x^18 + A)), n))};

Crossrefs

Cf. A128129, A128771.

Sequence in context: A090399 A069251 A261988 * A280947 A069252 A365667

Adjacent sequences: A128767 A128768 A128769 * A128771 A128772 A128773

Keyword

nonn

Author

Michael Somos, Mar 27 2007

Status

approved