%I A013973
%S A013973 1,504,16632,122976,532728,1575504,4058208,8471232,17047800,29883672,
%T A013973 51991632,81170208,129985632,187132176,279550656,384422976,545530104,715608432,
%U A013973 986161176,1247954400,1665307728,2066980608,2678616864,3243917376,4159663200
%V A013973 1,-504,-16632,-122976,-532728,-1575504,-4058208,-8471232,-17047800,-29883672,
%W A013973 -51991632,-81170208,-129985632,-187132176,-279550656,-384422976,-545530104,
               -715608432,
%X A013973 -986161176,-1247954400,-1665307728,-2066980608,-2678616864,-3243917376,
               -4159663200
%N A013973 Expansion of Eisenstein series E_6(q) (alternate convention E_3(q)).
%D A013973 D. Bump, Automorphic Forms..., Cambridge Univ. Press, 1997, p. 29.
%D A013973 W. Ebeling, Lattices and Codes, Vieweg; 2nd ed., 2002, see p. 53.
%D A013973 R. C. Gunning, Lectures on Modular Forms. Princeton Univ. Press, Princeton, 
               NJ, 1962, p. 53.
%D A013973 N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Springer-Verlag, 
               1984, see p. 111.
%D A013973 Jean-Pierre Serre, "A Course in Arithmetic", Springer, 1978
%D A013973 Joseph H. Silverman, "Advanced Topics in the Arithmetic of Elliptic Curves", 
               Springer, 1994
%D A013973 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.
%H A013973 T. D. Noe, <a href="b013973.txt">Table of n, a(n) for n=0..1000</a>
%H A013973 R. E. Borcherds, <a href="http://www.math.berkeley.edu/~reb/papers/">
               Automorphic forms on O_{s+2,2}(R)^{+} and generalized Kac-Moody algebras</
               a>, pp. 744-752 of Proc. Intern. Congr. Math., Vol. 2, 1994.
%H A013973 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 A013973 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 A013973 <a href="Sindx_Ed.html#Eisen">Index entries for sequences related to 
               Eisenstein series</a>
%F A013973 E6(q) = 1 - 504 sum_{i=1}^infinity sigma_5(i)q^i where sigma_5(n) is 
               A001160, the sum of fifth powers of the divisors of n. It can also 
               be expressed as E6(q) = 1 - 504 sum_{i=1}^infinity i^5 q^i/(1-q^i). 
               - Gene Ward Smith (genewardsmith(AT)gmail.com), Aug 22 2006
%F A013973 G.f. A(x) satisfies 0=f(A(x), A(x^2), A(x^4)) where f(u, v, w)= u^2*v 
               -8*u^2*w -66*u*v^2 +592*u*v*w -512*u*w^2 +121*v^3 -4224*v^2*w +4096*v*w^2. 
               - Michael Somos, Apr 10 2005
%F A013973 Expansion of Ramanujan's function R(q)= 216*g3 (Weierstrass invariant).
%F A013973 Expansion of (eta(q)^8 + 32 * eta(q^4)^8) * (eta(q)^16 - 512 * eta(q)^8 
               * eta(q^4)^8 - 8192 * eta(q^4)^16) / eta(q^2)^12 in powers of q. 
               - Michael Somos Dec 30 2008
%F A013973 G.f. is a period 1 Fourier series which satisfies f(-1 / t) = - (t/i)^6 
               * f(t) where q = exp(2 pi i t). - Michael Somos Dec 30 2008
%e A013973 1 - 504*q - 16632*q^2 - 122976*q^3 - 532728*q^4 - 1575504*q^5 + ...
%p A013973 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(6);
%o A013973 (PARI) a(n)=if(n<1,n==0,-504*sigma(n,5))
%o A013973 (PARI) {a(n) = local(A, A1, A4); if( n<0, 0, A = x * O(x^n); A1 = eta(x 
               + A)^8; A4 = 32 * x * eta(x^4 + A)^8; polcoeff( (A1 + A4) * (A1^2 
               - 16 * A1 * A4 - 8 * A4^2) / eta(x^2 + A)^12, n))} /* Michael Somos 
               Dec 30 2008 */
%Y A013973 Cf. A004009, A008410, A013974, A145095.
%Y A013973 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 A013973 Cf. A001160.
%Y A013973 Sequence in context: A141145 A166763 A012829 this_sequence A012744 A145095 
               A035293
%Y A013973 Adjacent sequences: A013970 A013971 A013972 this_sequence A013974 A013975 
               A013976
%K A013973 sign,easy
%O A013973 0,2
%A A013973 N. J. A. Sloane (njas(AT)research.att.com).