A106204 - OEIS

%I #17 Mar 12 2021 22:24:43

%S 1,0,2,-1,-14,30,140,-434,-1370,6579,13020,-100040,-101611,1500338,

%T 245954,-22069601,14502792,316451640,-480024439,-4385787620,

%U 10970363300,57983545059,-217649312794,-714104478148,3986473537118,7776402179076

%N Expansion of (chi(-q^3)^8 + 16*q^2/ chi(-q^3)^8)^(1/8) in powers of q where chi() is a Ramanujan theta function.

%C Ramanujan theta functions: f(q) := Product_{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} q^(k*(k+1)/2) (A010054), chi(q) := Product_{k>=0} (1+q^(2k+1)) (A000700).

%H G. C. Greubel, <a href="/A106204/b106204.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^(1/8)*((eta(q^3)/ eta(q^6))^8 + 16*(eta(q^6)/ eta(q^3))^8)^(1/8) in powers of q.

%t eta[q_]:= q^(1/24)*QPochhammer[q]; a[n_] := SeriesCoefficient[q^(1/8)*((eta[q^3]/eta[q^6])^8 + 16*(eta[q^6]/eta[q^3])^8)^(1/8), {q, 0, n}]; Table[a[n], {n, 0, 50}] (* _G. C. Greubel_, Mar 07 2018 *)

%o (PARI) {a(n)=local(A); if(n<0,0, A=x*O(x^n); A=(eta(x^3+A)/eta(x^6+A))^8; polcoeff( (A+16*x^2/A)^(1/8),n))}

%Y Cf. A000122, A000700, A007263, A010054, A121373.

%K sign

%O 0,3

%A _Michael Somos_, Apr 25 2005