A100004 - OEIS

%I #18 Apr 26 2021 13:04:17

%S 1,0,0,0,1260230400,0,211691822284800,167823813692620800,

%T 478913396715566841600,800774658647826785894400,

%U 611331534352048834410577920,247750040775246869910100377600,59524143410351572596837132825600,9219467698757233237143045380505600,982518868151763901368740969501491200

%N Theta series of Barnes-Wall lattice in 128 dimensions.

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

%H Andy Huchala, <a href="/A100004/b100004.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Ba#BW">Index entries for sequences related to Barnes-Wall lattices</a>

%e Theta series = 1+1260230400*q^8+211691822284800*q^12+167823813692620800*q^14+

%e 478913396715566841600*q^16+800774658647826785894400*q^18+

%e 611331534352048834410577920*q^20+247750040775246869910100377600*q^22+

%e 59524143410351572596837132825600*q^24+9219467698757233237143045380505600*q^26+

%e 982518868151763901368740969501491200*q^28+

%e 75861944796267943358105372399071395840*q^30+

%e 4424187851601897335090905879137082310400*q^32+

%e 201630718820389821054181008698644837171200*q^34+

%e 7386564136116987592917746020874575301836800*q^36+

%e 222716047981438421360112524163657218025062400*q^38+O(q^40)

%e or in terms of e8 and the cusp form:

%e e8^16-3840*e8^13*delta+4942080*e8^10*delta^2-2416128000*e8^7*delta^3

%e +364791398400*e8^4*delta^4-8259045949440*e8*delta^5.

%o (Sage) e8 = eisenstein_series_qexp(4,25,normalization = "integral");

%o delta = CuspForms(1,12).0.q_expansion(25);

%o f = e8^16-3840*e8^13*delta+4942080*e8^10*delta^2-2416128000*e8^7*delta^3+364791398400*e8^4*delta^4-8259045949440*e8*delta^5;

%o f[:25]

%K nonn

%O 0,5

%A Eric Rains and _N. J. A. Sloane_, Sep 27 2005