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A058550 Eisenstein series E_14(q) (alternate convention E_7(q)).

%I A058550 #15 Aug 03 2025 02:47:46
%S A058550 1,-24,-196632,-38263776,-1610809368,-29296875024,-313495116768,
%T A058550 -2325336249792,-13195750342680,-61004818143672,-240029297071632,
%U A058550 -828545091454368,-2568152034827232,-7269002558214096,-19051479894545856,-46708710975763776
%N A058550 Eisenstein series E_14(q) (alternate convention E_7(q)).
%D A058550 R. C. Gunning, Lectures on Modular Forms. Princeton Univ. Press, Princeton, NJ, 1962, p. 53.
%D A058550 N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Springer-Verlag, 1984, see p. 111.
%H A058550 Seiichi Manyama, <a href="/A058550/b058550.txt">Table of n, a(n) for n = 0..10000</a>
%H A058550 <a href="/index/Ed#Eisen">Index entries for sequences related to Eisenstein series</a>
%p A058550 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(14);
%t A058550 terms = 16;
%t A058550 E14[x_] = 1 - 24*Sum[k^13*x^k/(1 - x^k), {k, 1, terms}];
%t A058550 E14[x] + O[x]^terms // CoefficientList[#, x]&
%t A058550 (* or: *)
%t A058550 Table[If[n == 0, 1, -24*DivisorSigma[13, n]], {n, 0, terms-1}] (* _Jean-François Alcover_, Feb 26 2018 *)
%t A058550 (* or *)
%t A058550 terms = 15; E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}]; E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}]; CoefficientList[Series[E4[x]^2*E6[x], {x, 0, terms}], x] (* _Vaclav Kotesovec_, Aug 03 2025 *)
%o A058550 (PARI) a(n)=if(n<1,n==0,-24*sigma(n,13))
%Y A058550 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).
%K A058550 sign
%O A058550 0,2
%A A058550 _N. J. A. Sloane_, Dec 25 2000