A013974 Eisenstein series E_10(q) (alternate convention E_5(q)).

1, -264, -135432, -5196576, -69341448, -515625264, -2665843488, -10653352512, -35502821640, -102284205672, -264515760432, -622498190688, -1364917062432, -2799587834736, -5465169838656, -10149567696576, -18177444679944

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.

Index entries for sequences related to 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

Cf. A008410.

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).

Convolution of A004009 and A013973.

Sequence in context: A239183 A289062 A294181 * A289024 A145639 A285836

Adjacent sequences: A013971 A013972 A013973 * A013975 A013976 A013977

Keyword

sign

Author

N. J. A. Sloane

Status

approved