A128770 - OEIS

%I #16 Mar 12 2021 22:24:44

%S 1,2,4,8,14,24,40,64,100,152,228,336,488,700,992,1392,1934,2664,3640,

%T 4936,6648,8896,11832,15648,20584,26942,35096,45512,58768,75576,96816,

%U 123568,157156,199200,251676,316992,398072,498460,622448,775216,963012

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

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

%H G. C. Greubel, <a href="/A128770/b128770.txt">Table of n, a(n) for n = 0..1000</a>

%H Vaclav Kotesovec, <a href="http://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], 2015-2016.

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

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

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

%F 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).

%F 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).

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

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

%F a(n) ~ exp(2*Pi*sqrt(2*n)/3) / (2^(3/4) * 3^(3/2) * n^(3/4)). - _Vaclav Kotesovec_, Oct 13 2015

%e 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 + ...

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q^9] / EllipticTheta[ 4, 0, q], {q, 0, n}]; (* _Michael Somos_, Apr 26 2015 *)

%t 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 *)

%o (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))};

%Y Cf. A128129, A128771.

%K nonn

%O 0,2

%A _Michael Somos_, Mar 27 2007