OFFSET
0,2
REFERENCES
W. Ebeling, Lattices and Codes, Vieweg; 2nd ed., 2002, see p. 53.
R. C. Gunning, Lectures on Modular Forms. Princeton Univ. Press, Princeton, NJ, 1962, p. 53.
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.
N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Springer-Verlag, 1984, see p. 111.
Jean-Pierre Serre, "A Course in Arithmetic", Springer, 1978
Joseph H. Silverman, "Advanced Topics in the Arithmetic of Elliptic Curves", Springer, 1994
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
R. E. Borcherds, Automorphic forms on O_{s+2,2}(R)^{+} and generalized Kac-Moody algebras, pp. 744-752 of Proc. Intern. Congr. Math., Vol. 2, 1994.
D. Bump, Automorphic Forms and Representations, Cambr. Univ. Press, 1997, p. 29.
Heng Huat Chan, Shaun Cooper, and Pee Choon Toh, Ramanujan's Eisenstein series and powers of Dedekind's eta-function, Journal of the London Mathematical Society 75.1 (2007): 225-242. See R(q).
Masao Koike, Modular forms on non-compact arithmetic triangle groups, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021. I wrote 2005 on the first page but the internal evidence suggests 1997.]
H. Ochiai, Counting functions for branched covers of elliptic curves and quasi-modular forms, arXiv:math-ph/9909023, 1999.
Eric Weisstein's World of Mathematics, Eisenstein Series.
FORMULA
E6(q) = 1 - 504*Sum_{i>=1} 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} i^5*q^i/(1-q^i). - Gene Ward Smith, Aug 22 2006
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
Expansion of Ramanujan's function R(q) = 216*g3 (Weierstrass invariant).
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
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
E6(q) = eta(q)^24 / eta(q^2)^12 - 480 * eta(q^2)^12 - 16896 * eta(q^2)^12 * eta(q^4)^8 / eta(q)^8 + 8192 * eta(q^4)^24 / eta(q^2)^12. - Seiichi Manyama, May 08 2017
EXAMPLE
G.f. = 1 - 504*q - 16632*q^2 - 122976*q^3 - 532728*q^4 - 1575504*q^5 + ...
MAPLE
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);
# alternative
A013973 := proc(n)
if n = 0 then
1;
else
-504*numtheory[sigma][5](n) ;
end if;
end proc:
seq(A013973(n), n=0..10) ; # R. J. Mathar, Feb 22 2021
MATHEMATICA
a[ n_] := If[ n < 1, Boole[n == 0], -504 DivisorSigma[ 5, n]]; (* Michael Somos, Apr 21 2013 *)
a[ n_] := SeriesCoefficient[ With[ {t2 = EllipticTheta[ 2, 0, q]^4, t3 = EllipticTheta[ 3, 0, q]^4}, t2^3 - 33 (t2 + t3) t2 t3 + t3^3], {q, 0, n}]; (* Michael Somos, Apr 21 2013 *)
a[ n_] := SeriesCoefficient[ With[ {t3 = EllipticTheta[ 3, 0, q]^4, t4 = EllipticTheta[ 4, 0, q]^4}, (t3^3 - 3 (t3 - t4)^2 (t3 + t4) + t4^3) / 2], {q, 0, 2 n}]; (* Michael Somos, Jun 04 2014 *)
a[ n_] := SeriesCoefficient[ With[ {e1 = QPochhammer[ q]^8, e4 = 32 q QPochhammer[ q^4]^8}, (e1 + e4) (e1^2 - 16 e1 e4 - 8 e4^2) / QPochhammer[ q^2]^12], {q, 0, n}]; (* Michael Somos, Apr 01 2015 *)
a[ n_] := SeriesCoefficient[ With[ {t2 = EllipticTheta[ 2, 0, q]^4, t3 = EllipticTheta[ 3, 0, q]^4}, t2^3 - 3/2 (t2 + t3) t2 t3 + t3^3], {q, 0, 2 n}]; (* Michael Somos, Jul 31 2016 *)
terms = 25; E6[x_] = 1-(12/BernoulliB[6])*Sum[k^5*x^k/(1-x^k), {k, terms}]; CoefficientList[E6[x] + O[x]^terms, x] (* Jean-François Alcover, Feb 28 2018 *)
PROG
(PARI) {a(n) = if( n<1, n==0, -504 * sigma( n, 5))};
(PARI) {a(n) = my(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 */
(Sage) ModularForms( Gamma1(1), 6, prec=25).0; # Michael Somos, Jun 04 2013
(Magma) Basis( ModularForms( Gamma1(1), 6), 25); /* Michael Somos, Apr 01 2015 */
CROSSREFS
KEYWORD
sign,easy
AUTHOR
STATUS
approved