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A195734 G.f.: exp( Sum_{n>=1} (2*sigma(n^2) - sigma(n)^2) * x^n/n ).

%I A195734 #13 Oct 31 2024 12:17:23
%S A195734 1,1,3,6,11,22,40,72,123,215,363,605,991,1618,2598,4139,6537,10229,
%T A195734 15871,24476,37487,56995,86177,129531,193662,287992,426254,627841,
%U A195734 920708,1344331,1954987,2831688,4086168,5875087,8417724,12020250,17108958,24275947,34340966
%N A195734 G.f.: exp( Sum_{n>=1} (2*sigma(n^2) - sigma(n)^2) * x^n/n ).
%H A195734 Vaclav Kotesovec, <a href="/A195734/b195734.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Paul D. Hanna)
%F A195734 Logarithmic derivative equals A195735.
%F A195734 log(a(n)) ~ 3*(5*zeta(3)*(12 - Pi^2))^(1/3) * n^(2/3) / (2*Pi^(2/3)). - _Vaclav Kotesovec_, Oct 31 2024
%e A195734 G.f.: A(x) = 1 + x + 3*x^2 + 6*x^3 + 11*x^4 + 22*x^5 + 40*x^6 + 72*x^7 +...
%e A195734 where
%e A195734 log(A(x)) = x + 5*x^2/2 + 10*x^3/3 + 13*x^4/4 + 26*x^5/5 + 38*x^6/6 + 50*x^7/7 + 29*x^8/8 +...+ A195735(n)*x^n/n +...
%t A195734 nmax = 40; CoefficientList[Series[Exp[Sum[(2*DivisorSigma[1,k^2] - DivisorSigma[1,k]^2) * x^k / k, {k, 1, nmax}]], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Oct 31 2024 *)
%o A195734 (PARI) {a(n)=polcoeff(exp(sum(k=1, n,(2*sigma(k^2)-sigma(k)^2)*x^k/k)+x*O(x^n)), n)}
%Y A195734 Cf. A195735 (log), A156302, A156303, A000203.
%K A195734 nonn
%O A195734 0,3
%A A195734 _Paul D. Hanna_, Sep 22 2011