%I A004011 M5140
%S A004011 1,24,24,96,24,144,96,192,24,312,144,288,96,336,192,576,24,432,312,480,
%T A004011 144,768,288,576,96,744,336,960,192,720,576,768,24,1152,432,1152,312,
%U A004011 912,480,1344,144,1008,768,1056,288,1872,576,1152,96,1368,744,1728,336
%N A004011 Theta series of D_4 lattice; Fourier coefficients of Eisenstein series 
               E_{gamma,2}.
%C A004011 D_4 is also the Barnes-Wall lattice in 4 dimensions.
%C A004011 E_{gamma,2} is the unique normalized modular form for Gamma_0(2) of weight 
               2.
%D A004011 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A004011 N. J. A. Sloane, Seven Staggering Sequences, in Homage to a Pied Puzzler, 
               E. Pegg Jr., A. H. Schoen and T. Rodgers (editors), A. K. Peters, 
               Wellesley, MA, 2009, pp. 93-110.
%D A004011 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", 
               Springer-Verlag, p. 119.
%D A004011 H. Cohn, Advanced Number Theory, Dover Publications, Inc., 1980, p. 89. 
               Eq. (1).
%H A004011 T. D. Noe, <a href="b004011.txt">Table of n, a(n) for n = 0..10000</a>
%H A004011 N. Heninger, E. M. Rains and N. J. A. Sloane, <a href="http://arXiv.org/
               abs/math.NT/0509316">On the Integrality of n-th Roots of Generating 
               Functions</a>, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
%H A004011 B. Brent, <a href="http://www.expmath.org/expmath/volumes/7/7.html">Quadratic 
               Minima and Modular Forms, Experimental Mathematics, v.7 no.3, 257-274.</
               a>
%H A004011 Michael Gilleland, <a href="selfsimilar.html">Some Self-Similar Integer 
               Sequences</a>
%H A004011 G. Nebe and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/
               lattices/D4.html">Home page for D_4 lattice</a>
%H A004011 N. J. A. Sloane, <a href="a004011.gif">The 24 minimal vectors form the 
               24-cell polytope</a>
%H A004011 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/g4g7.pdf">
               Seven Staggering Sequences</a>.
%H A004011 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               24-Cell.html">Link to a section of The World of Mathematics.</a>
%H A004011 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               EisensteinSeries.html">Link to a section of The World of Mathematics.</
               a>
%H A004011 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A004011 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A004011 <a href="Sindx_Da.html#D4">Index entries for sequences related to D_4 
               lattice</a>
%H A004011 <a href="Sindx_Ed.html#Eisen">Index entries for sequences related to 
               Eisenstein series</a>
%H A004011 <a href="Sindx_Ba.html#BW">Index entries for sequences related to Barnes-Wall 
               lattices</a>
%H A004011 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Barnes-WallLattice.html">Barnes-Wall Lattice</a>
%F A004011 a(0)=1; if n>0 then a(n)=24 (sum_{d|n, d odd, d>0} d).
%F A004011 G.f.: 1+24 Sum_{n>0} nx^n/(1+x^n).
%F A004011 G.f. A(x) satisfies 0=f(A(x), A(x^2), A(x^4)) where f(u, v, w)=u^2-2*u*v-7*v^2-8*v*w+16*w^2 
               . - Michael Somos May 29 2005
%F A004011 Expansion of (1+k^2)K(k^2)^2/(pi/2)^2 in powers of nome q. - Michael 
               Somos Jun 10 2006
%F A004011 G.f.: (1/2)*(theta_3(z)^4 + theta_4(z)^4) = theta_3(2z)^4 + theta_2(2z)^4 
               = Sum_{k>=0} a(k)x^(2k).
%F A004011 G.f. is a period 1 Fourier series which satisfies f(-1 / (2 t)) = 2 (t/
               i)^2 f(t) where q = exp(2 pi i t). - Michael Somos Sep 11 2007
%F A004011 G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, 
               u3, u6) = u1^2 +4*u2^2 +9*u3^2 +36*u6^2 -2*u1*u2 -10*u1*u3 +10*u1*u6 
               +10*u2*u3 -40*u2*u6 -18*u3*u6. - Michael Somos Sep 11 2007
%e A004011 1 + 24*q^2 + 24*q^4 + 96*q^6 + 24*q^8 + 144*q^10 + 96*q^12 + 192*q^14 
               + 24*q^16 + ...
%p A004011 readlib(ifactors): with(numtheory): for n from 1 to 100 do if n mod 2 
               = 0 then m := n/ifactors(n)[2][1][1]^ifactors(n)[2][1][2] else m 
               := n fi: printf(`%d,`,24*sigma(m)) od: # from James A. Sellers Dec 
               07 2000
%o A004011 (PARI) a(n)=if(n<1,n==0,24*sumdiv(n,d,d%2*d))
%o A004011 (PARI) {a(n) = if( n<1, n==0, qfrep([ 2,1,1,1; 1,2,0,0; 1,0,2,0; 1,0,
               0,2], n, 1)[n] * 2 )} /* Michael Somos Sep 11 2007 */
%Y A004011 a(n)=24*A000593(n), n>0. Partial sums give A046949. Cf. A108092, A108096.
%Y A004011 A000118(2n)=A096727(2n)=a(n).
%Y A004011 Cf. A108092 (fourth root).
%Y A004011 Sequence in context: A022358 A122505 A103640 this_sequence A056465 A056455 
               A128378
%Y A004011 Adjacent sequences: A004008 A004009 A004010 this_sequence A004012 A004013 
               A004014
%K A004011 nonn,easy,core,nice
%O A004011 0,2
%A A004011 N. J. A. Sloane (njas(AT)research.att.com).
%E A004011 Additional comments from Barry Brent (barryb(AT)primenet.com)