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%I #36 Sep 08 2022 08:44:48
%S 1,0,10,20,0,20,0,60,50,0,30,0,60,120,0,60,0,160,70,0,60,0,120,220,0,
%T 120,0,200,180,0,40,0,210,240,0,180,0,360,200,0,150,0,120,420,0,140,0,
%U 460,220,0,130,0,360,520,0,240,0,400,300,0,180,0,320,420,0,360,0,660,480,0,120
%N Theta series of A*_4 lattice.
%D J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 114.
%H John Cannon, <a href="/A023916/b023916.txt">Table of n, a(n) for n = 0..5000</a>
%H S. Ahlgren, <a href="http://dx.doi.org/10.1090/S0002-9939-99-05181-3">The sixth, eighth, ninth and tenth powers of Ramanujan's theta function</a>, Proc. Amer. Math. Soc., 128 (1999), 1333-1338; F_5(q).
%H G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/As4.html">Home page for this lattice</a>
%F Expansion of f(-x)^5 / f(-x^5) + 5 * x * f(-x^5)^5 / f(-x) in powers of x where f() is a Ramanujan theta function. - _Michael Somos_, Jan 29 2011
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (5 t)) = 5^(1/2) (t/i)^2 g(t) where q = exp(2*Pi*i*t) and g() is g.f. for A008444. - _Michael Somos_, Jan 29 2011
%F a(5*n) = A008444(n). a(5*n + 1) = a(5*n + 4) = 0. - _Michael Somos_, Jan 29 2011
%e 1 + 10*x^2 + 20*x^3 + 20*x^5 + 60*x^7 + 50*x^8 + 30*x^10 + 60*x^12 + ...
%e 1 + 10*q^4 + 20*q^6 + 20*q^10 + 60*q^14 + 50*q^16 + 30*q^20 + 60*q^24 + 120*q^26 + 60*q^30 + 160*q^34 + 70*q^36 + 60*q^40 + 120*q^44 + 220*q^46 + 120*q^50 + 200*q^54 + 180*q^56 + 40*q^60 + O(q^62).
%t a[n_] := Module[{A = x*O[x]^n}, SeriesCoefficient[QPochhammer[x+A]^5 / QPochhammer[x^5+A] + 5*x*(QPochhammer[x^5+A]^5 / QPochhammer[x+A]), {x, 0, n}]]; Table[a[n], {n, 0, 80}] (* _Jean-François Alcover_, Nov 05 2015, adapted from PARI *)
%o (Magma) L:=Lattice("A",4); D:=Dual(L); T1<q> := ThetaSeries(D,60);
%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^5 / eta(x^5 + A) + 5 * x * eta(x^5 + A)^5 / eta(x + A), n))}; /* _Michael Somos_, Jan 29 2011 */
%Y Cf. A008444.
%K nonn
%O 0,3
%A _N. J. A. Sloane_.
%E More terms from _N. J. A. Sloane_, Dec 24 2006
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