%I A004009 M5416
%S A004009 1,240,2160,6720,17520,30240,60480,82560,140400,181680,272160,319680,
%T A004009 490560,527520,743040,846720,1123440,1179360,1635120,1646400,2207520,
%U A004009 2311680,2877120,2920320,3931200,3780240,4747680,4905600,6026880
%N A004009 Expansion of Eisenstein series E_4(q) (alternate convention E_2(q)); 
               theta series of E_8 lattice.
%C A004009 E_8 is also the Barnes-Wall lattice in 8 dimensions.
%C A004009 Expansion of Ramanujan's function Q(q)=12g2 (Weierstrass invariant).
%D A004009 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A004009 D. Bump, Automorphic Forms..., Camb., 1997 p. 29.
%D A004009 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", 
               Springer-Verlag, p. 123.
%D A004009 H. S. M. Coxeter, Integral Cayley numbers, Duke Math. J. 13 (1946), 561-578; 
               reprinted in "Twelve Geometric Essays", pp. 20-39.
%D A004009 W. Ebeling, Lattices and Codes, Vieweg; 2nd ed., 2002, see p. 53.
%D A004009 R. C. Gunning, Lectures on Modular Forms. Princeton Univ. Press, Princeton, 
               NJ, 1962, p. 53.
%D A004009 M. Kaneko and D. Zagier, Supersingular j-invariants, hypergeometric series 
               and Atkin's orthogonal polynomials, pp. 97-126 of D. A. Buell and 
               J. T. Teitelbaum, eds., Computational Perspectives on Number Theory, 
               Amer. Math. Soc., 1998.
%D A004009 N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Springer-Verlag, 
               1984, see p. 111.
%D A004009 S. Ramanujan, On Certain Arithmetical Functions, Messenger Math., 45 
               (1916), 11-15 (Eq. (25)). Collected Papers of Srinivasa Ramanujan, 
               Chap. 16, Ed. G. H. Hardy et al., Chelsea, NY, 1962.
%D A004009 S. Ramanujan, On Certain Arithmetical Functions, Messenger Math., 45 
               (1916), 11-15 (Eq. (25)). Ramanujan's Papers, p. 196, Ed. B. J. Venkatachala 
               et al., Prism Books, Bangalore 2000.
%D A004009 Jean-Pierre Serre, "A Course in Arithmetic", Springer, 1978
%D A004009 Joseph H. Silverman, "Advanced Topics in the Arithmetic of Elliptic Curves", 
               Springer, 1994
%D A004009 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.
%H A004009 N. J. A. Sloane, <a href="b004009.txt">Table of n, a(n) for n = 0..1000</
               a>
%H A004009 H. H. Chan and C. Krattenthaler, <a href="http://arXiv.org/abs/math.NT/
               0407061">Recent progress in the study of representations of integers 
               as sums of squares</a>
%H A004009 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 A004009 G. Nebe and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/
               lattices/E8.html">Home page for E_8 lattice</a>
%H A004009 H. Ochiai, <a href="http://arXiv.org/abs/math-ph/9909023">Counting functions 
               for branched covers of elliptic curves and quasi-modular forms</a>
%H A004009 S. Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/
               Cpaper37/page1.htm">On the coefficients in the expansions of certain 
               modular functions</a>, Proc. Royal Soc., A, 95 (1918), 144-155.
%H A004009 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/sg.txt">
               My favorite integer sequences</a>, in Sequences and their Applications 
               (Proceedings of SETA '98).
%H A004009 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/g4g7.pdf">
               Seven Staggering Sequences</a>.
%H A004009 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 A004009 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               LeechLattice.html">Link to a section of The World of Mathematics.</
               a>
%H A004009 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Barnes-WallLattice.html">Barnes-Wall Lattice</a>
%H A004009 <a href="Sindx_Ed.html#Eisen">Index entries for sequences related to 
               Eisenstein series</a>
%H A004009 <a href="Sindx_Ba.html#BW">Index entries for sequences related to Barnes-Wall 
               lattices</a>
%F A004009 Can also be expressed as E4(q) = 1 + 240 sum_{i=1}^infinity i^3 q^i/(1-q^i) 
               - Gene Ward Smith (genewardsmith(AT)gmail.com), Aug 22 2006
%F A004009 1 + 240*Sum ( sigma_3 (m) * q^2m ), m = 1..inf, where sigma_3 (m) is 
               the sum of the cubes of the divisors of m (A001158).
%F A004009 G.f. A(x) satisfies 0=f(A(x), A(x^2), A(x^4)) where f(u, v, w)=u^2+33*v^2+256*w^2-18*u*v+16*u*w-288*v*w 
               . - Michael Somos Jan 05 2006
%F A004009 Expansion of (phi(-q)^8 -(2phi(-q)phi(q))^4 +16phi(q)^8) in powers of 
               q where phi() is a Ramanujan theta function.
%F A004009 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 +16*u2^2 +81*u3^2 +1296*u6^2 -14*u1*u2 -18*u1*u3 +30*u1*u6 
               +30*u2*u3 -288*u2*u6 -1134*u3*u6 . - Michael Somos Apr 15 2007
%F A004009 G.f. A(x) satisfies 0=f(A(x), A(x^3), A(x^9)) where f(u, v, w)= +u^3*v 
               +9*w*u^3 -84*u^2*v^2 +246*u*v^3 -253*v^4 -675*w*u^2*v +729*w^2*u^2 
               -4590*w*u*v^2 +19926*w*v^3 -54675*w^2*u*v +59049*w^3*u +531441*w^3*v 
               -551124*w^2*v^2 . - Michael Somos Apr 15 2007
%F A004009 Expansion of (eta(q)^24 + 256 * eta(q^2)^24) / (eta(q) * eta(q^2))^8 
               in powers of q. - Michael Somos Dec 30 2008
%F A004009 G.f. is a period 1 Fourier series which satisfies f(-1 / t) = (t/i)^4 
               * f(t) where q = exp(2 pi i t). - Michael Somos Dec 30 2008
%e A004009 1 + 240*q + 2160*q^2 + 6720*q^3 + 17520*q^4 + 30240*q^5 + 60480*q^6 + 
               ...
%p A004009 with(numtheory); E := proc(k) local n,t1; t1 := 1-(2*k/bernoulli(k))*add(sigma[k-1](n)*q^n,
               n=1..60); series(t1,q,60); end; E(4);
%o A004009 (PARI) a(n)=if(n<1,n==0,240*sigma(n,3))
%o A004009 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x 
               + A)^24 + 256 * x * eta(x^2 + A)^24) / (eta(x + A) * eta(x^2 + A))^8, 
               n))} /* Michael Somos Dec 30 2008 */
%Y A004009 Cf. A046948, A000143, A108091 (eighth root).
%Y A004009 Cf. A001158.
%Y A004009 A008410 is convolution of this sequence with itself. A008411 is convolution 
               of this sequence with A008410.
%Y A004009 Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6), A008410 (E_8), A013974 
               (E_10), A029828 (E_12), A058550 (E_14), A029829 (E_16), A029830 (E_20), 
               A029831 (E_24).
%Y A004009 Sequence in context: A158562 A157766 A049335 this_sequence A005950 A004536 
               A145094
%Y A004009 Adjacent sequences: A004006 A004007 A004008 this_sequence A004010 A004011 
               A004012
%K A004009 nonn,easy,nice
%O A004009 0,2
%A A004009 N. J. A. Sloane (njas(AT)research.att.com).