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%I #9 Jan 10 2019 23:08:39
%S 1,3,9,45,153,702,2754,11259,50058,224046,990873,4304988,18175185,
%T 77439321,327135510,1408297995,6302244447,29140976502,135628856406,
%U 622855827801,2796140431278,12364271615628,54378167114070,240937280782164,1080881256295566
%N Expansion of g.f.: exp( Sum_{n>=1} 3^n*(Sum_{d|n} d*x^d)^n/n ).
%H G. C. Greubel, <a href="/A192891/b192891.txt">Table of n, a(n) for n = 0..1000</a>
%e G.f.: A(x) = 1 + 3*x + 9*x^2 + 45*x^3 + 153*x^4 + 702*x^5 + 2754*x^6 + 11259*x^7 +...
%e where the logarithm of the g.f. begins:
%e log(A(x)) = 3*x + 9*(x + 2*x^2)^2/2 + 27*(x + 3*x^3)^3/3 + 81*(x + 2*x^2 + 4*x^4)^4/4 + 243*(x + 5*x^5)^5/5 + 729*(x + 2*x^2 + 3*x^3 + 6*x^6)^6/6 + 2187*(x + 7*x^7)^7/7 + 6561*(x + 2*x^2 + 4*x^4 + 8*x^8)^8/8 +...
%e Explicitly, the logarithmic series begins:
%e log(A(x)) = 3*x + 9*x^2/2 + 81*x^3/3 + 153*x^4/4 + 1458*x^5/5 + 3645*x^6/6 + 20898*x^7/7 + 100521*x^8/8 + 557685*x^9/9 + 2353374*x^10/10 +...
%t With[{m = 40}, CoefficientList[Series[Exp[Sum[3^n (Sum[d*x^d, {d, Divisors[n]}])^n/n, {n, 1, m + 2}]], {x, 0, m}], x]] (* _G. C. Greubel_, Jan 10 2019 *)
%o (PARI) {a(n)=local(A);A=exp(sum(m=1,n+1,3^m*sumdiv(m,d,d*x^d +x*O(x^n))^m/m));polcoeff(A,n)}
%Y Cf. A192860, A192890.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jul 11 2011
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