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%I A056945 #10 May 18 2022 16:59:33
%S A056945 1,0,0,-4,6,0,0,32736,131076,0,0,3669012,9172952,0,0,95691552,
%T A056945 188239518,0,0,1142929524,1959705000,0,0,8506686816,13293227112,0,0,
%U A056945 45763087664,67073100864,0,0,195387947712,272567759508,0,0,698077783656,938807478318,0,0,2176654050912
%N A056945 Jacobi form of weight 12 and index 1 associated to a (nonexistent) lattice vector of norm 2 for the Leech lattice.
%C A056945 Let J(h)=E_8*E_{4,1}+(2h-60)*phi_{12,1} be the Jacobi form of weight 12 and index 1 associated with a norm 2 vector of a Niemeier lattice of Coxeter number h. Let J(h)=sum_{n,r} c(4n-r^2) q^n*z^r. So a(n)=c(4m-r^2) for h=0.
%C A056945 Let N(h,n) be the number of vectors of norm 2n for the lattice, then we have N(h,n)=c(4n)+2*sum_{1<=r<=sqrt(4n)}c(4n-r^2) if h is the Coxeter number of a Niemeier lattice. Note that N(0,n)=a(4n)-2*sum a(4n-r^2)=A008408(n), for the Leech lattice! Note also a(3)<0 and a(n) is nonnegative for n<=1000, except 3.
%D A056945 Eichler and Zagier, The Theory of Jacobi Forms, Birkhauser, 1985.
%F A056945 E_8*E_{4, 1}-60*phi_{12, 1}. The E's are Eisenstein-Jacobi series and phi_{12, 1} is the unique normalized Jacobi cusp form of weight 12 and index 1.
%F A056945 a(n) = A055747(n) - 60*A003785(n). - _Sean A. Irvine_, May 18 2022
%Y A056945 Cf. A003785, A008408, A056946, A055747.
%K A056945 sign
%O A056945 0,4
%A A056945 Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jul 16 2000

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