A037150 - OEIS

%I #43 Dec 12 2020 23:56:26

%S 1,0,-1512,44352,449064,6519744,47263608,257027904,1115041032,

%T 4093040448,13000566096,37057027392,96945887304,232758852480,

%U 526296318912,1128198915648,2286101175624,4451375005056,8386154766360,15131349955008,26614555499952

%N Fourier coefficients of Eisenstein series of degree 2 and weight 6 when evaluated at Gram(A_2)*z.

%D J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, Third Ed., 1998.

%D Helmut Klingen, Introductory Lectures on Siegel Modular Forms, Cambrifge, 1990, p. 123.

%H N. J. A. Sloane, <a href="/A037150/b037150.txt">Table of n, a(n) for n = 0..100</a>

%H N. J. A. Sloane, <a href="/A037150/a037150.pdf">Notes on Two-dimensional Theta Series of Lattices</a> (Notes on some joint work with Eric M. Rains), pages 96-115, circa Jun 08 1998, of N. J. A. Sloane's notebook "Lattices Volume 79".

%H <a href="/index/Ed#Eisen">Index entries for sequences related to Eisenstein series</a>

%F x^12-72*x^6*y-1728*y^2, x = phi_0(z), y = Delta_12(z). Here phi_0(z) is the theta series of the A_2 lattice - see A004016, and Delta_12 is the 12-dimensional cusp form given in A007332.

%p # Maple code from _N. J. A. Sloane_, Dec 12 2020. Will also be useful for related sequences.

%p # get standard theta series in Maple. First set max degree, maxd.

%p maxd:=201:

%p # get th2, th3, th4 = Jacobi theta constants out to degree maxd (Ref. Conway-Sloane, p. 102)

%p temp0:=trunc(evalf(sqrt(maxd)))+2:

%p a:=0: for i from -temp0 to temp0 do a:=a+q^( (i+1/2)^2): od:

%p th2:=series(a,q,maxd);

%p a:=0: for i from -temp0 to temp0 do a:=a+q^(i^2): od:

%p th3:=series(a,q,maxd);

%p th4:=series(subs(q=-q,th3),q,maxd);

%p # get Dedekind eta function

%p a:=q^(1/24) : for m from 1 to maxd do a:=a*(1-q^m); od:

%p eta:=a;

%p # get phi0 and phi1 (Ref. Conway-Sloane, p. 103)

%p phi0:=series( subs(q=q^2,th2)*subs(q=q^6,th2)+subs(q=q^2,th3)*subs(q=q^6,th3), q, maxd );

%p 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 # get delta12 (Ref. Conway-Sloane, p. 204, where it is called Delta_6)

%p delta12:=series((subs(q=q^3,eta)*eta)^6,q,maxd);

%p delta12:=series(subs(q=q^2,delta12),q,maxd);

%p # To get the present sequence: (Ref. Sloane notebook pages)

%p x:=phi0; y:=delta12;

%p w1:= x^12-72*x^6*y-1728*y^2; w1s:=series(w1,q,maxd); w2:=subs(q=sqrt(t),w1s); w3:=series(w2,t,101);

%p w4:=seriestolist(w3); # A037150

%Y Cf. A004016, A007332.

%K sign

%O 0,3

%A _N. J. A. Sloane_.

%E Entry revised by _N. J. A. Sloane_, Dec 12 2020