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%I #25 Mar 12 2021 22:24:41
%S 1,0,0,0,10,32,0,0,40,0,0,0,80,160,0,0,90,0,0,0,112,320,0,0,240,0,0,0,
%T 320,480,0,0,200,0,0,0,250,800,0,0,560,0,0,0,560,992,0,0,400,0,0,0,
%U 560,1120,0,0,800,0,0,0,960,1760,0,0,730,0,0,0,480,1920,0,0,1240,0,0,0,1520,1920
%N Theta series of D*_5 lattice.
%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).
%D J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 120, Eq. 96.
%H N. J. A. Sloane, <a href="/A008422/b008422.txt">Table of n, a(n) for n = 0..800</a>
%H G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/Ds5.html">Home page for this lattice</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 Theta series in terms of Jacobi theta series: (theta_2)^5 + (theta_3)^5.
%F Expansion of phi(q^4)^5 + ( 2 * q * psi(q^8) )^5 in powers of q where phi(), psi() are Ramanujan theta functions. - _Michael Somos_, Sep 17 2007
%t terms = 78; phi[q_] := EllipticTheta[3, 0, q]; chi[q_] := ((1 - InverseEllipticNomeQ[q])*InverseEllipticNomeQ[q]/(16*q))^(-1/24); psi[q_] := (1/2)*q^(-1/8)*EllipticTheta[2, 0, q^(1/2)]; s = phi[q^4]^5 + (2*q*psi[q^8])^5 + O[q]^terms; CoefficientList[s, q] (* _Jean-François Alcover_, Jul 04 2017, after _Michael Somos_ *)
%o (PARI)
%o N=66; q='q+O('q^N);
%o T3(q) = eta(q^2)^5 / ( eta(q)^2 * eta(q^4)^2 );
%o T2(q) = eta(q^4)^2 / eta(q^2);
%o Vec( T3(q^4)^5 + (2 * q * T2(q^4))^5 )
%o \\ _Joerg Arndt_, Mar 30 2018
%K nonn,easy
%O 0,5
%A _N. J. A. Sloane_
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