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%I A209397 #14 Feb 15 2020 23:16:10
%S A209397 1,3,7,19,46,129,337,939,2581,7238,20263,57337,162319,461961,1317217,
%T A209397 3767035,10792400,30983565,89084845,256531814,739658815,2135234247,
%U A209397 6170505666,17849457873,51679366171,149750711581,434260829464,1260198317509,3659410074933
%N A209397 L.g.f.: Sum_{n>=1} a(n)*x^n/n  =  Sum_{n>=1} x^n/n * exp( Sum_{k>=1} a(k)*x^(n*k)/k ).
%H A209397 Alois P. Heinz, <a href="/A209397/b209397.txt">Table of n, a(n) for n = 1..2136</a> (first 500 terms from Paul D. Hanna)
%F A209397 a(n) = Sum_{d|n} d*A000081(d).
%F A209397 L.g.f.: Sum_{n>=1} -A000081(n) * log(1-x^n).
%F A209397 L.g.f.: log( G(x)/x ) = Sum_{n>=1} G(x^n)/n where G(x) is the g.f. of A000081, which is the number of rooted trees with n nodes.
%F A209397 a(n) ~ c * d^n / sqrt(n), where d = A051491 = 2.9557652856519949747148..., c = A187770 = 0.4399240125710253040409... . - _Vaclav Kotesovec_, Oct 30 2014
%e A209397 L.g.f.: L(x) = x + 3*x^2/2 + 7*x^3/3 + 19*x^4/4 + 46*x^5/5 + 129*x^6/6 +...
%e A209397 Let G(x) be the g.f. of A000081, then
%e A209397 exp(L(x)) = G(x)/x where G(x) = x*exp( Sum_{n>=1} G(x^n)/n ) begins:
%e A209397 G(x) = x + x^2 + 2*x^3 + 4*x^4 + 9*x^5 + 20*x^6 + 48*x^7 + 115*x^8 + 286*x^9 + 719*x^10 + 1842*x^11 + 4766*x^12 + 12486*x^13 + 32973*x^14 +...
%o A209397 (PARI) {a(n)=local(L=vector(n, i, 1)); for(i=1, n, L=Vec(deriv(sum(m=1, n, x^m/m*exp(sum(k=1, n\m, L[k]*x^(m*k)/k)+x*O(x^n)))))); L[n]}
%o A209397 for(n=1,30,print1(a(n),","))
%Y A209397 Cf. A000081, A203253.
%K A209397 nonn
%O A209397 1,2
%A A209397 _Paul D. Hanna_, Mar 07 2012

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