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%I M5309 #33 Sep 08 2022 08:44:33
%S 1,0,54,72,0,432,270,0,918,720,0,2160,936,0,2700,2160,0,5184,2214,0,
%T 5616,3600,0,9504,4590,0,9180,6552,0,15120,5184,0,14742,10800,0,21600,
%U 9360,0,19548,12240,0,30240,13500,0,28080,17712,0,39744,14760,0,32454
%N Theta series of {E_6}* lattice.
%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%D J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 127.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H N. Heninger, E. M. Rains and N. J. A. Sloane, <a href="http://arXiv.org/abs/math.NT/0509316">On the Integrality of n-th Roots of Generating Functions</a>, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
%H G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/Es6.html">Home page for this lattice</a>
%F Expansion of b(q)^3 + c(q)^3 / 3 in power of q where b(), c() are cubic AGM theta functions.
%F Expansion of (eta(q)^3 / eta(q^3))^3 + 9 * (eta(q^3)^3 / eta(q))^3 in powers of q.
%e G.f. = 1 + 54*x^2 + 72*x^3 + 432*x^5 + 270*x^6 + 918*x^8 + 720*x^9 + 2160*x^11 + ...
%t a[n_] := Module[{A = x*O[x]^n}, SeriesCoefficient[(QPochhammer[x+A]^3 / QPochhammer[x^3+A])^3 + 9*x*(QPochhammer[x^3+A]^3 / QPochhammer[x+A])^3, {x, 0, n}]]; Table[a[n], {n, 0, 80}] (* _Jean-François Alcover_, Nov 05 2015, adapted from 1st PARI script *)
%t a[ n_] := SeriesCoefficient[ (QPochhammer[ q]^3 / QPochhammer[ q^3])^3 + 9 q (QPochhammer[ q^3]^3 /QPochhammer[ q])^3 , {q, 0, n}]; ( _Michael Somos_, Dec 28 2015 )
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^3 / eta(x^3 + A))^3 + 9 * x * (eta(x^3 + A)^3 / eta(x + A))^3, n))}; /* _Michael Somos_, Feb 28 2012 */
%o (PARI) {a(n) = my(A, a1, p3); if( n<0, 0, A = x * O(x^n); a1 = sum( k=1, n, 6 * sumdiv(k, d, kronecker( d, 3)) * x^k, 1 + A); p3 = sum( k=1, n\3, -24 * sigma(k) * x^(3*k), 1 + A); polcoeff( (a1^3 + a1 * p3 - 4 * x * a1') / 2, n))}; /* _Michael Somos_, Feb 28 2012 */
%o (Magma) A := Basis( ModularForms( Gamma1(3), 3), 51); A[1]; /* _Michael Somos_, Dec 28 2015 */
%Y Cf. A004007 (E_6).
%K nonn,nice,easy
%O 0,3
%A _N. J. A. Sloane_