A124243 - OEIS

%I #10 Nov 16 2017 10:39:54

%S 1,-1,1,-2,3,-4,5,-7,10,-12,15,-20,26,-32,39,-50,63,-76,92,-114,140,

%T -168,201,-244,295,-350,415,-496,591,-696,818,-967,1140,-1332,1554,

%U -1820,2126,-2468,2861,-3324,3855,-4448,5126,-5916,6816,-7824,8970,-10292,11793,-13471,15372,-17548

%N Expansion of q*psi(q^9)/psi(q) in powers of q.

%H G. C. Greubel, <a href="/A124243/b124243.txt">Table of n, a(n) for n = 1..1000</a>

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

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

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

%F G.f.: x*Product_{k>0} (1-x^k)*(1-x^(18k))^2/((1-x^(2k))^2*(1-x^(9k))).

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

%t eta[x_] := x^(1/24)*QPochhammer[x]; A124243[n_] := SeriesCoefficient[ (eta[q]/eta[q^9])*(eta[q^18]/eta[q^2])^2, {q, 0, n}]; Table[A124243[n], {n, 0, 50}] (* _G. C. Greubel_, Aug 26 2017 *)

%o (PARI) {a(n)=local(A); if(n<1, 0, n--; A=x*O(x^n); polcoeff( eta(x+A)*eta(x^18+A)^2/eta(x^2+A)^2/eta(x^9+A), n))}

%K sign

%O 1,4

%A _Michael Somos_, Oct 28 2006