A123759 - OEIS

%I #13 Mar 12 2021 22:24:44

%S 1,-1,-1,0,0,0,1,2,0,0,-1,0,-2,0,0,-2,1,2,0,0,0,0,0,0,0,0,1,0,0,0,2,

%T -2,-1,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,2,0,0,0,0,2,0,2,0,0,0,0,-2,

%U 0,0,-2,-1,0,0,0,-2,0,0,0,0,0,2,0,0,0,-1,1,0,0,0,0,0,2,0,0,0,0,2,0,0,0,0,-2,0,0,0,0,0,0,0

%N Expansion of f(-q)*psi(-q^5) in powers of q where f(), psi() are Ramanujan theta functions.

%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="/A123759/b123759.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 Euler transform of period 20 sequence [ -1, -1, -1, -1, -2, -1, -1, -1, -1, -1, -1, -1, -1, -1, -2, -1, -1, -1, -1, -2, ...].

%F Product_{k>0} (1-x^k)*(1-x^(5k))*(1+x^(10k)).

%F a(8n+3) = a(8n+5) = 0.

%F Expansion of q^(-2/3) * eta(q) * eta(q^5) * eta(q^20)/ eta(q^10) in powers of q.

%t eta[q_]:= q^(1/24)*QPochhammer[q]; a[n_]:= SeriesCoefficient[q^(-2/3)* eta[q]*eta[q^5]*eta[q^20]/eta[q^10], {q, 0, n}]; Table[a[n], {n, 0, 50}] (* _G. C. Greubel_, Mar 08 2018 *)

%o (PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x+A)*eta(x^5+A)*eta(x^20+A)/eta(x^10+A), n))}

%o (PARI) {a(n) = local(s, k); if(n<0, 0, n=24*n+16; forstep(k=1, sqrtint(n\15),2, if(issquare(n-15*k^2, &j)& (j^2%6==1), s+= (-1)^((j+1)\6+ (k+2)\4))); s)}

%K sign

%O 0,8

%A _Michael Somos_, Oct 12 2006