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Numerators of coefficients of Eisenstein series E_12(q) (or E_6(q) or E_24(q)).
2

%I #26 Aug 06 2024 04:42:26

%S 1,65520,134250480,11606736960,274945048560,3199218815520,

%T 23782204031040,129554448266880,563087459516400,2056098632318640,

%U 6555199353000480,18693620658498240,48705965462306880,117422349017369760,265457064498837120,566735214731736960,1153203117089652720

%N Numerators of coefficients of Eisenstein series E_12(q) (or E_6(q) or E_24(q)).

%C First denominator is 1, rest are 691.

%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.

%D J.-P. Serre, Course in Arithmetic, Chap. VII, Section 4.

%H Andy Huchala, <a href="/A037164/b037164.txt">Table of n, a(n) for n = 0..20000</a>

%H H. D. Nguyen, D. Taggart, <a href="https://citeseerx.ist.psu.edu/pdf/8f2f36f22878c984775ed04368b8893879b99458">Mining the OEIS: Ten Experimental Conjectures</a>, 2013; Mentions this sequence. - From _N. J. A. Sloane_, Mar 16 2014

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

%p with(numtheory):

%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(12);

%p seq(numer(coeff(%,q,n)), n=0..24);

%t terms = 13; E12[x_] = 1 - (24/BernoulliB[12])*Sum[k^11*x^k/(1 - x^k), {k, 1, terms}]; E12[x] + O[x]^terms // CoefficientList[#, x]& // Numerator (* _Jean-François Alcover_, Feb 27 2018 *)

%o (Sage)

%o l = list(eisenstein_series_qexp(12,20, normalization='integral'))

%o l[0] = 1; l # _Andy Huchala_, Jul 01 2021

%Y Cf. A029828.

%K nonn,frac,easy

%O 0,2

%A _N. J. A. Sloane_.