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%I #25 Mar 12 2021 22:24:42
%S 1,2,-1,-2,0,-4,-1,2,-4,2,4,2,1,-2,4,2,4,0,-4,0,-3,4,-4,-4,0,-2,0,-6,
%T 0,2,-1,-4,4,-4,-4,8,4,6,0,2,-8,0,7,2,4,2,4,0,0,-6,4,0,-4,0,0,0,1,-6,
%U -4,4,-8,-2,-4,4,0,2,-4,-6,0,-2,4,-8,1,2,0,0,4,4,4,-2,4,6,0,-2,0,-4,-8,10,8,8,-1,4,4,2,-4,-4,-8,6,4,-6,8,-6,4,4
%N Expansion of theta_4(q^2) * theta_2(q)^2/(4*q^(1/2)) in powers of q.
%C The nonzero quadrisection of A248395.
%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="/A080966/b080966.txt">Table of n, a(n) for n = 0..5000</a>
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H J. B. Tunnell, <a href="http://dx.doi.org/10.1007/BF01389327">A classical Diophantine problem and modular forms of weight 3/2</a>, Invent. Math., 72 (1983), 323-334.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F G.f.: Product_{k>0} (1+x^k)^2*(1-x^(2k))^3/(1+x^(2k)).
%F Expansion of f(-q^4)*f(q)^2 in powers of q where f(-q)=f(-q,-q^2) is a Ramanujan theta function.
%F Expansion of q^(-1/4)*eta(q^2)^6/(eta(q)^2*eta(q^4)) in powers of q.
%F Euler transform of period-4 sequence [2,-4,2,-3,...].
%F G.f.: Product_{k>0} (1-x^(2*k))^3*(1+x^k)^2/(1+x^(2*k)).
%F 2*a(n) = A080964(4*n+1) = 2*A072071(4*n+1) - A072070(4*n+1).
%e q + 2*q^5 - q^9 - 2*q^13 - 4*q^21 - q^25 + 2*q^29 - 4*q^33 + ...
%t QP = QPochhammer; s = QP[q^2]^6/(QP[q]^2*QP[q^4]) + O[q]^100; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 14 2015, adapted from PARI *)
%t QP := QPochhammer; a:=CoefficientList[Series[QP[q^2]^6/(QP[q]^2*QP[q^4]), {q, 0, 60}], q]; Table[a[[n]], {n, 1, 50}] (* _G. C. Greubel_, Jul 01 2018 *)
%o (PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^2+A)^6/eta(x+A)^2/eta(x^4+A), n))}
%K sign
%O 0,2
%A _Michael Somos_, Feb 28 2003
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