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A080963 Expansion of theta_3(q)*theta_3(q^2)*theta_4(q^8) in powers of q. 4

%I #17 Mar 12 2021 22:24:42

%S 1,2,2,4,2,0,4,0,0,2,-4,-4,0,0,-8,0,-2,-8,6,-4,-8,0,4,0,0,-6,-12,0,0,

%T 0,-8,0,-4,8,8,-8,10,0,12,0,0,0,-8,12,0,0,-8,0,8,2,14,8,-8,0,16,0,0,8,

%U -4,4,0,0,-16,0,6,0,16,-4,16,0,8,0,0,8,-20,-4,0,0,-8,0,-8,-6,8,4,-16,0,20,0,0,-8,-20,-8,0,0,-16,0,-8

%N Expansion of theta_3(q)*theta_3(q^2)*theta_4(q^8) 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="/A080963/b080963.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 a(16*n+5) = a(16*n+7) = a(16*n+8) = a(16*n+12) = a(16*n+13) = a(16*n+15) = 0.

%F a(n) = 2*A080918(n) - A080917(n).

%F a(2*n+1) = 2*A034950(n).

%F Expansion of (eta(q^2)*eta(q^4))^3/(eta(q)^2*eta(q^16)) in powers of q.

%F Euler transform of period-16 sequence [2,-1,2,-4,2,-1,2,-4,2,-1,2,-4,2,-1,2,-3,...].

%F Expansion of phi(q)phi(q^2)phi(-q^8) in powers of q where phi() is a Ramanujan theta function.

%F G.f.: Product_{k>0} (1+x^k)^2*(1-x^(2k))*(1-x^(4k))^2/((1+x^(4k))*(1+x^(8k))). - _Michael Somos_, Feb 16 2006

%t a[n_]:= SeriesCoefficient[EllipticTheta[3,0,q]*EllipticTheta[3,0,q^2]* EllipticTheta[3,0,-q^8], {q,0,n}]; Table[a[n], {n,0,50}] (* or *)

%t eta[q_] := q^(1/24)*QPochhammer[q]; a[n_]:= SeriesCoefficient[ (eta[q^2]*eta[q^4])^3/(eta[q]^2*eta[q^16]), {q,0,n}]; Table[a[n], {n,0,50}] (* _G. C. Greubel_, Feb 11 2018 *)

%o (PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( (eta(x^2+A)*eta(x^4+A))^3/(eta(x+A)^2*eta(x^16+A)), n))}

%K sign

%O 0,2

%A _Michael Somos_, Feb 28 2003

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Last modified March 29 08:13 EDT 2024. Contains 371265 sequences. (Running on oeis4.)