cp's OEIS Frontend

A004010 Theta series of 12-dimensional Coxeter-Todd lattice K_12.

%I A004010 M5478 #39 Feb 16 2025 08:32:27
%S A004010 1,0,756,4032,20412,60480,139860,326592,652428,1020096,2000376,
%T A004010 3132864,4445532,7185024,10747296,13148352,21003948,27506304,33724404,
%U A004010 48009024,64049832,70709184,102958128,124782336,142254252,189423360,237588120,248250240,344391264
%N A004010 Theta series of 12-dimensional Coxeter-Todd lattice K_12.
%C A004010 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%D A004010 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 129.
%D A004010 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A004010 Seiichi Manyama, <a href="/A004010/b004010.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..500 from N. J. A. Sloane)
%H A004010 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 A004010 G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/K12.html">Home page for this lattice</a>
%H A004010 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Coxeter-ToddLattice.html">Coxeter-Todd Lattice</a>
%H A004010 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ThetaSeries.html">Theta Series</a>
%F A004010 G.f. is a period 1 Fourier series which satisfies f(-1 / (3 t)) = 27 (t/i)^6 f(t) where q = exp(2 Pi i t). - _Michael Somos_, Dec 20 2015
%F A004010 G.f.: (3*a(x)^6 - 4*a(x)^3*b(x)^3 + 4*b(x)^6) / 3 where a(), b() are cubic AGM theta functions. - _Michael Somos_, Dec 25 2015
%e A004010 G.f. = 1 + 756*x^2 + 4032*x^3 + 20412*x^4 + 604890*x^5 + 139860*x^6 + ...
%e A004010 G.f. = 1 + 756*q^4 + 4032*q^6 + 20412*q^8 + 60480*q^10 + 139860*q^12 + 326592*q^14 + 652428*q^16 + 1020096*q^18 + 2000376*q^20 + ...
%p A004010 # Jacobi theta constants th2, th3: maxd := 201: temp0 := trunc(evalf(sqrt(maxd)))+2: a := 0: for i from -temp0 to temp0 do a := a+q^( (i+1/2)^2): od: th2 := series(a,q,maxd); a := 0: for i from -temp0 to temp0 do a := a+q^(i^2): od: th3 := series(a,q,maxd);
%p A004010 # get phi0 and phi1: phi0 := series( subs(q=q^2, th2)*subs(q=q^6, th2)+subs(q=q^2, th3)*subs(q=q^6, th3), q, maxd); phi1 := series( subs(q=q^2, th2)*subs(q=q^6, th3)+subs(q=q^2, th3)*subs(q=q^6, th2), q, maxd);
%p A004010 K_12 := series( subs(q=q^2,phi0)^6+45*subs(q=q^2,phi0)^2*subs(q=q^2,phi1)^4+18*subs(q=q^2,phi1)^6,q,maxd);
%t A004010 maxd = 51; temp0 = Floor[ Sqrt[maxd] ]+2; a = 0; Do[ a=a+q^(i+1/2)^2, {i, -temp0, temp0}]; th2[q_] = Normal[ Series[a, {q, 0, maxd}]]; a = 0; Do[ a=a+q^i^2, {i, -temp0, temp0}]; th3[q_] = Normal[ Series[a, {q, 0, maxd}]]; phi0[q_] = Normal[ Series[ th2[q^2]*th2[q^6] + th3[q^2]*th3[q^6], {q, 0, maxd}]]; phi1[q_] = Normal[ Series[ th2[q^2]*th3[q^6] + th3[q^2]*th2[q^6], {q, 0, maxd}]]; K12 = Series[ phi0[q^2]^6 + 45*phi0[q^2]^2*phi1[q^2]^4 + 18*phi1[q^2]^6, {q, 0, maxd}]; CoefficientList[ K12, q^2 ] (* _Jean-François Alcover_, Nov 28 2011, translated from Maple *)
%t A004010 a[ n_] := With[ {U1 = QPochhammer[ q]^3, U3 = QPochhammer[ q^3]^3, U9 = QPochhammer[ q^9]^3}, With[ {z = (1 + 9 q U9/U1)^3}, SeriesCoefficient[ (U1^3/U3)^2 (3 z^2 - 4 z + 4) / 3, {q, 0, n}]]]; (* _Michael Somos_, Dec 25 2015 *)
%o A004010 (Magma) A := Basis( ModularForms( Gamma1(3), 6), 29); A[1] + 756*A[3]; /* _Michael Somos_, Dec 20 2015 */
%o A004010 (PARI) {a(n) = my(A, U1, U3, U9, z); if( n<0, 0, A = x * O(x^n); U1 = eta(x + A)^3; U3 = eta(x^3 + A)^3; U9 = eta(x^9 + A)^3; z = (1 + 9 * x * U9/U1)^3; polcoeff( (U1^3/U3)^2 * (3*z^2 - 4*z +4) / 3, n))}; /* _Michael Somos_, Dec 25 2015 */
%Y A004010 Cf. A004046, A008657, A107658.
%K A004010 easy,nonn,nice
%O A004010 0,3
%A A004010 _N. J. A. Sloane_