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A123758 Expansion of q^(-1/3)*eta(q)*eta(q^4)*eta(q^5)/eta(q^2) in powers of q. 1

%I

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

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

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

%N Expansion of q^(-1/3)*eta(q)*eta(q^4)*eta(q^5)/eta(q^2) in powers of q.

%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..oo} q^(k*(k+1)/2) (A010054), chi(q) := Product_{k>=0} (1+q^(2k+1)) (A000700).

%H G. C. Greubel, <a href="/A123758/b123758.txt">Table of n, a(n) for n = 0..1000</a>

%H M. Somos, <a href="http://somos.crg4.com/multiq.pdf">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 f(-q^5)*psi(-q) in powers of q where f(),psi() are Ramanujan theta functions.

%F Euler transform of period 20 sequence [ -1, 0, -1, -1, -2, 0, -1, -1, -1, -1, -1, -1, -1, 0, -2, -1, -1, 0, -1, -2, ...].

%F Product_{k>0} (1-x^k)*(1+x^(2k))*(1-x^(5k)).

%F a(8n+2) = a(8n+4) = 0.

%t eta[q_] := q^(1/24)*QPochhammer[q]; CoefficientList[Series[q^(-1/3) eta[q] eta[q^4] eta[q^5]/eta[q^2], {q, 0, 100}], q] (* _G. C. Greubel_, Apr 19 2018 *)

%o (PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x+A)*eta(x^4+A)*eta(x^5+A)/eta(x^2+A), n))}

%o (PARI) {a(n) = local(s, k); if(n<0, 0, n=24*n+8; for(j=1, sqrtint(n\5), if((j^2%6==1)& issquare( (n-5*j^2)/3, &k)& (k%2), s+= (-1)^((j+1)\6+ (k+2)\4))); s)}

%K sign

%O 0,7

%A _Michael Somos_, Oct 12 2006

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Last modified July 24 02:42 EDT 2019. Contains 325287 sequences. (Running on oeis4.)