Search: id:A004020
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%I A004020 M0931
%S A004020 2,4,2,4,4,0,6,4,0,4,4,4,2,4,0,4,8,0,4,0,2,8,4,0,4,4,0,4,4,4,2,8,0,0,4,
%T A004020 0,8,4,4,4,0,0,6,4,0,4,8,0,4,4,0,8,0,0,0,8,6,4,4,0,4,4,0,0,4,4,8,4,0,4,
%U A004020 4,0,6,4,0,0,8,0,4,4,0,12,0,4,4,0,0,4,4,0,2,8,4,4,8,0,0,4,0,4,4,4,4,0
%N A004020 Theta series of square lattice with respect to edge.
%C A004020 Number of solutions in integers of n = x^2+y^2+y.
%D A004020 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", 
               Springer-Verlag, p. 106.
%D A004020 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%F A004020 G.f.: 2(Sum_{k>=0} x^((k^2+k)/2))^2 = (Sum_k x^(k^2+k))(Sum_k x^(k^2)).
%F A004020 Expansion of q^(-1/2)c(q)/2 in powers of q^2 where c(q) is the third 
               function in the quadratic Gauss AGM. - Michael Somos, Feb 10 2006
%F A004020 Expansion of 2 * phi(q) * psi(q^2) in powers of q where phi(), psi() 
               are Ramanujan theta functions. - Michael Somos, Feb 10 2006
%o A004020 (PARI) a(n)=local(X); if(n<0,0,X=x+x*O(x^n); 2*polcoeff(eta(X^2)^4/eta(X)^2,
               n))
%o A004020 (PARI) a(n)=2*if(n<1,n==0,polcoeff(sum(k=0,(sqrtint(8*n+1)-1)\2,x^(k*(k+1)/
               2),x*O(x^n))^2,n))
%Y A004020 a(n)=2*A008441(n)=A004531(4n+1).
%Y A004020 Sequence in context: A163894 A032059 A074075 this_sequence A143235 A069465 
               A047947
%Y A004020 Adjacent sequences: A004017 A004018 A004019 this_sequence A004021 A004022 
               A004023
%K A004020 nonn
%O A004020 0,1
%A A004020 N. J. A. Sloane (njas(AT)research.att.com).

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