OFFSET
0,2
REFERENCES
R. C. Gunning, Lectures on Modular Forms. Princeton Univ. Press, Princeton, NJ, 1962, p. 53.
N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Springer-Verlag, 1984, see p. 111.
LINKS
T. D. Noe, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Eisenstein Series.
FORMULA
Sum_{n >= 0} a(n)/exp(Pi)^(2n) = 0 or is very close to 0. - Gerald McGarvey, Jan 25 2005
G.f. is a period 1 Fourier series which satisfies f(-1 / t) = - (t/i)^10 * f(t) where q = exp(2 Pi i t). - Michael Somos, Dec 30 2008
G.f.: 1 - 264*Sum_{k>=1} k^9*x^k/(1 - x^k). - Ilya Gutkovskiy, Aug 31 2017
EXAMPLE
G.f. = 1 - 264*q - 135432*q^2 - 5196576*q^3 - 69341448*q^4 - 515625264*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(10);
MATHEMATICA
a[ n_] := If[ n < 1, Boole[n == 0], -264 DivisorSigma[ 9, n]]; (* Michael Somos, Jun 04 2013 *)
a[ n_] := SeriesCoefficient[ With[{t2 = EllipticTheta[ 2, 0, q]^4, t3 = EllipticTheta[ 3, 0, q]^4}, t2^5 - 19 t2 t3 (t2^3 + t3^3) - 494 (t2 t3)^2 (t2 + t3) + t3^5], {q, 0, n}]; (* Michael Somos, Jun 04 2013 *)
terms = 17; Ei[n_] = 1-(2n/BernoulliB[n]) Sum[k^(n-1) x^k/(1-x^k), {k, terms}]; CoefficientList[Ei[10] + O[x]^terms, x] (* Jean-François Alcover, Mar 01 2018 *)
PROG
(PARI) {a(n) = if( n<1, n==0, -264 * sigma( n, 9))};
(Sage) ModularForms( Gamma1(1), 10, prec=13).0; # Michael Somos, Jun 04 2013
CROSSREFS
KEYWORD
sign
AUTHOR
STATUS
approved