A139135 - OEIS

%I #13 Mar 12 2021 22:24:45

%S 1,-1,2,-4,6,-9,14,-20,29,-42,58,-80,110,-148,198,-264,347,-454,592,

%T -764,982,-1257,1598,-2024,2554,-3206,4010,-5000,6208,-7684,9484,

%U -11664,14306,-17501,21346,-25972,31526,-38170,46112,-55588,66861,-80258,96154,-114968,137212

%N Expansion of psi(-q^3) / f(q) where psi(), f() are Ramanujan theta functions.

%C Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

%H G. C. Greubel, <a href="/A139135/b139135.txt">Table of n, a(n) for n = 0..1000</a>

%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 q^(-1/3) * eta(q) * eta(q^3) * eta(q^4) * eta(q^12) / (eta(q^2)^3 * eta(q^6)) in powers of q.

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (108 t)) = 3^(-1/2) g(t) where q = exp(2 Pi i t) and g() is g.f. for A139136.

%F a(n) ~ (-1)^n * exp(Pi*sqrt(2*n/3)) / (2^(9/4) * 3^(3/4) * n^(3/4)). - _Vaclav Kotesovec_, Nov 16 2017

%e q - q^4 + 2*q^7 - 4*q^10 + 6*q^13 - 9*q^16 + 14*q^19 - 20*q^22 + 29*q^25 + ...

%t A139135[n_] := SeriesCoefficient[(QPochhammer[q]* QPochhammer[q^3]*QPochhammer[q^4]*QPochhammer[q^12])/(QPochhammer[q^2]^3 *QPochhammer[q^6]), {q, 0, n}]; Table[A139135[n], {n, 0, 50}] (* _G. C. Greubel_, Oct 05 2017 *)

%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A) / (eta(x^2 + A)^3 * eta(x^6 + A)), n))}

%Y A139136(3*n + 1) = - a(n). A139137(3*n + 1) = 2 * a(n).

%Y Apart from signs, same as A097197.

%K sign

%O 0,3

%A _Michael Somos_, Apr 10 2008