Search: id:A030212
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%I A030212
%S A030212 0,1,4,0,16,14,0,0,64,81,56,0,0,238,0,0,256,322,324,0,224,0,0,0,0,429,
%T A030212 952,0,0,82,0,0,1024,0,1288,0,1296,2162,0,0,896,3038,0,0,0,1134,0,0,0,
%U A030212 2401,1716,0,3808,2482,0,0,0,0,328,0,0,6958,0,0,4096,3332,0,0,5152,0,0
%V A030212 0,1,-4,0,16,-14,0,0,-64,81,56,0,0,-238,0,0,256,322,-324,0,-224,0,0,0,
               0,-429,952,0,0,
%W A030212 82,0,0,-1024,0,-1288,0,1296,2162,0,0,896,-3038,0,0,0,-1134,0,0,0,2401,
               1716,0,-3808,
%X A030212 2482,0,0,0,0,-328,0,0,-6958,0,0,4096,3332,0,0,5152,0,0
%N A030212 Expansion of eta(q)^4*eta(q^2)^2*eta(q^4)^4.
%C A030212 Euler transform of period 4 sequence [ -4,-6,-4,-10,...]. - Michael Somos, 
               Jul 17 2004
%C A030212 Multiplicative with a(2^e)=(-4)^e, a(p^e)=p^2e if e even else 0 for p=3 
               mod 4.
%C A030212 Called chi_4(n) by Glaisher and Hardy.
%D A030212 G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his 
               life and work, Chelsea Publishing Company, New York 1959, p. 135 
               section 9.3. MR0106147 (21 #4881)
%D A030212 M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.
%D A030212 H. McKean and V. Moll, Elliptic Curves, Cambridge University Press, 1997, 
               page 175, 4.7 Exercise 5. MR1471703 (98g:14032)
%D A030212 J. W. L. Glaisher, On the representation of a number as sum of 2,4,6,
               8... squares, Quart. J. Math. 38 (1907), 1-62 (see p. 34).
%F A030212 G.f. x(Product_{k>0} (1-x^k)^4(1-x^(2k))^2(1-x^(4k))^4). a(4k+3)=0. a(4k)=16a(k).
%F A030212 G.f.: (t*t''-3(t')^2)/2 where t=theta_3(x) (A000122) and t' := x*(dt/
               dx), t'' := (t')'. - Michael Somos Nov 08 2005
%F A030212 Given A=A0+A1+A2+A3 is the 4-section, then 0=(A0^2-A2^2)^2+4*A0*A2*A1^2 
               . - Michael Somos Mar 08 2006
%e A030212 q -4*q^2 +16*q^4 -14*q^5 -64*q^8 +81*q^9 +56*q^10 -238*q^13 +...
%o A030212 (PARI) a(n)=local(A); if(n<1,0, n--; A=x^n*O(x); polcoeff((eta(x+A)*eta(x^4+A))^4*eta(x^2+A)^2,
               n)) /* Michael Somos, Jul 17 2004 */
%o A030212 (PARI) {a(n)=local(r); if(n<1, 0, r=sqrtint(n); sum(x=-r,r, sum(y=-r,
               r, if(x^2+y^2==n, (x+I*y)^4) ))/4 )} /* Michael Somos Sep 12 2005 
               */
%Y A030212 Sequence in context: A167361 A167314 A158802 this_sequence A167359 A007216 
               A057378
%Y A030212 Adjacent sequences: A030209 A030210 A030211 this_sequence A030213 A030214 
               A030215
%K A030212 sign,mult
%O A030212 0,3
%A A030212 N. J. A. Sloane (njas(AT)research.att.com).

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