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%I #13 Mar 31 2018 08:25:38
%S 1,1,3,9,20,52,105,253,536,1142,2421,4999,10278,20686,41512,81984,
%T 161029,312681,603070,1153284,2189331,4129537,7733317,14399693,
%U 26644337,49034811,89741600,163411148,296074694,533909026,958416113,1712893825,3048468607,5403248469,9539609984
%N G.f.: exp( Sum_{n>=1} A068963(n)*x^n/n ) where A068963(n) = Sum_{d|n} phi(d^3).
%C Here phi(n) = A000010(n) is the Euler totient function.
%C Euler transform of A002618. - _Vaclav Kotesovec_, Mar 30 2018
%H Vaclav Kotesovec, <a href="/A226106/b226106.txt">Table of n, a(n) for n = 0..10000</a>
%H Vaclav Kotesovec, <a href="/A226106/a226106.jpg">Graph - The asymptotic ratio</a>
%F a(n) ~ exp(2^(9/4) * sqrt(Pi) * n^(3/4) / (3 * 5^(1/4)) + 3*Zeta(3) / Pi^2) / (2^(11/8) * 5^(1/8) * Pi^(1/4) * n^(5/8)). - _Vaclav Kotesovec_, Mar 30 2018
%e G.f.: A(x) = 1 + x + 3*x^2 + 9*x^3 + 20*x^4 + 52*x^5 + 105*x^6 + 253*x^7 +...
%e where
%e log(A(x)) = x + 5*x^2/2 + 19*x^3/3 + 37*x^4/4 + 101*x^5/5 + 95*x^6/6 + 295*x^7/7 + 293*x^8/8 + 505*x^9/9 +...+ A068963(n)*x^n/n +...
%t nmax = 40; CoefficientList[Series[Product[1/(1-x^k)^(k*EulerPhi[k]), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Mar 30 2018 *)
%t nmax = 40; CoefficientList[Series[Product[1/(1-x^k)^EulerPhi[k^2], {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Mar 30 2018 *)
%t nmax = 40; CoefficientList[Series[Exp[Sum[Sum[k*EulerPhi[k] * x^(j*k) / j, {k, 1, Floor[nmax/j] + 1}], {j, 1, nmax}]], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Mar 31 2018 *)
%o (PARI) {a(n)=polcoeff(exp(sum(m=1,n+1,sumdiv(m,d,eulerphi(d^3))*x^m/m)+x*O(x^n)),n)}
%o for(n=0,35,print1(a(n),", "))
%Y Cf. A068963, A002618, A061255, A299069, A301986.
%K nonn
%O 0,3
%A _Paul D. Hanna_, May 26 2013
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