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A283263 Expansion of exp( Sum_{n>=1} -sigma_3(n)*x^n/n ) in powers of x.

%I A283263 #27 Nov 11 2020 08:30:44
%S A283263 1,-1,-4,-5,-1,21,49,81,45,-121,-484,-997,-1344,-840,1624,6931,15149,
%T A283263 23155,23469,2240,-57596,-168929,-322587,-461165,-450668,-64135,
%U A283263 985621,2935044,5718865,8597971,9683008,5596899,-8414092,-37295629,-83336988,-141108721
%N A283263 Expansion of exp( Sum_{n>=1} -sigma_3(n)*x^n/n ) in powers of x.
%H A283263 Seiichi Manyama, <a href="/A283263/b283263.txt">Table of n, a(n) for n = 0..1000</a>
%F A283263 G.f.: Product_{n>=1} (1 - x^n)^(n^2).
%F A283263 a(n) = -(1/n)*Sum_{k=1..n} sigma_3(k)*a(n-k).
%t A283263 a[n_] := If[n<1, 1,-(1/n) * Sum[DivisorSigma[3, k] a[n - k], {k, n}]]; Table[a[n], {n, 0, 35}] (* _Indranil Ghosh_, Mar 16 2017 *)
%o A283263 (PARI) a(n) = if(n<1, 1, -(1/n) * sum(k=1, n, sigma(k, 3) * a(n - k)));
%o A283263 for(n=0, 35, print1(a(n), ", ")) \\ _Indranil Ghosh_, Mar 16 2017
%o A283263 (SageMath) # uses[EulerTransform from A166861]
%o A283263 b = EulerTransform(lambda n: -n^2)
%o A283263 print([b(n) for n in range(36)]) # _Peter Luschny_, Nov 11 2020
%Y A283263 Column k=2 of A283272.
%Y A283263 Cf. A023871 (exp( Sum_{n>=1} sigma_3(n)*x^n/n )).
%Y A283263 Cf. exp( Sum_{n>=1} -sigma_k(n)*x^n/n ): A010815 (k=1), A073592 (k=2), this sequence (k=3), A283264 (k=4), A283271 (k=5).
%K A283263 sign
%O A283263 0,3
%A A283263 _Seiichi Manyama_, Mar 04 2017