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A281374 Coefficients in q-expansion of E_2^2, where E_2 is the Eisenstein series shown in A006352.

%I A281374 #18 Aug 02 2025 14:12:43
%S A281374 1,-48,432,3264,9456,21600,39744,66432,105840,147984,220320,281664,
%T A281374 393792,475104,646272,743040,980592,1091232,1432944,1536960,1965600,
%U A281374 2118144,2649024,2761344,3516480,3557040,4433184,4594560,5575296,5603040,6998400,6864384,8407152,8494848,10085472,9918720,12319152
%N A281374 Coefficients in q-expansion of E_2^2, where E_2 is the Eisenstein series shown in A006352.
%H A281374 Seiichi Manyama, <a href="/A281374/b281374.txt">Table of n, a(n) for n = 0..1000</a>
%F A281374 For n>0, a(n) = 240*A001158(n) - 288*n*A000203(n). - _Vaclav Kotesovec_, Aug 02 2025
%p A281374 with(numtheory); M:=100;
%p A281374 E := proc(k) local n, t1; global M;
%p A281374 t1 := 1-(2*k/bernoulli(k))*add(sigma[k-1](n)*q^n, n=1..M+1);
%p A281374 series(t1, q, M+1); end;
%p A281374 e2:=E(2); e4:=E(4); e6:=E(6);
%p A281374 series(e2^2,q,M+1);
%p A281374 seriestolist(%);
%t A281374 terms = 37;
%t A281374 E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}];
%t A281374 E2[x]^2 + O[x]^terms // CoefficientList[#, x]& (* _Jean-François Alcover_, Feb 23 2018 *)
%t A281374 (* or *)
%t A281374 Join[{1}, Table[240*DivisorSigma[3, n] - 288*n*DivisorSigma[1, n], {n, 1, 50}]] (* _Vaclav Kotesovec_, Aug 02 2025 *)
%Y A281374 Cf. A000203, A001158, A006352.
%K A281374 sign
%O A281374 0,2
%A A281374 _N. J. A. Sloane_, Feb 05 2017