A013974 - OEIS

%I #34 Dec 27 2021 22:15:07

%S 1,-264,-135432,-5196576,-69341448,-515625264,-2665843488,

%T -10653352512,-35502821640,-102284205672,-264515760432,-622498190688,

%U -1364917062432,-2799587834736,-5465169838656,-10149567696576,-18177444679944

%N Eisenstein series E_10(q) (alternate convention E_5(q)).

%D R. C. Gunning, Lectures on Modular Forms. Princeton Univ. Press, Princeton, NJ, 1962, p. 53.

%D N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Springer-Verlag, 1984, see p. 111.

%H T. D. Noe, <a href="/A013974/b013974.txt">Table of n, a(n) for n = 0..1000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EisensteinSeries.html">Eisenstein Series.</a>

%H <a href="/index/Ed#Eisen">Index entries for sequences related to Eisenstein series</a>

%F Sum_{n >= 0} a(n)/exp(Pi)^(2n) = 0 or is very close to 0. - _Gerald McGarvey_, Jan 25 2005

%F 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

%F G.f.: 1 - 264*Sum_{k>=1} k^9*x^k/(1 - x^k). - _Ilya Gutkovskiy_, Aug 31 2017

%e G.f. = 1 - 264*q - 135432*q^2 - 5196576*q^3 - 69341448*q^4 - 515625264*q^5 + ...

%p 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);

%t a[ n_] := If[ n < 1, Boole[n == 0], -264 DivisorSigma[ 9, n]]; (* _Michael Somos_, Jun 04 2013 *)

%t 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 *)

%t 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 *)

%o (PARI) {a(n) = if( n<1, n==0, -264 * sigma( n, 9))};

%o (Sage) ModularForms( Gamma1(1), 10, prec=13).0; # _Michael Somos_, Jun 04 2013

%Y Cf. A008410.

%Y 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 Convolution of A004009 and A013973.

%K sign

%O 0,2

%A _N. J. A. Sloane_