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%I #7 Sep 17 2023 21:26:31
%S 1,2,12,120,2208,75840,5026048,654140416,168815832320,86777091183104,
%T 89034709122434048,182521862861195356160,747975313568170390573056,
%U 6128911186837999697172176896,100428344706090874604628656668672,3291036905110044354733349281915109376
%N G.f. satisfies: A(x) = x*exp( Sum_{n>=1} A(2^n*x^n) / n ).
%C Compare to: G(x) = x*exp( Sum_{n>=1} G(x^n)/n ), which is the g.f. of A000081, the number of rooted trees with n nodes.
%F Limit a(n) / 2^(n*(n-1)/2) = 2.4760521181770989525583758338042055853633514575492...
%e G.f.: A(x) = x + 2*x^2 + 12*x^3 + 120*x^4 + 2208*x^5 + 75840*x^6 + 5026048*x^7 + ...
%e where
%e A(x) = x*exp(A(2*x) + A(4*x^2)/2 + A(8*x^3)/3 + A(16*x^4)/4 + A(32*x^5)/5 + A(64*x^6)/6 + A(128*x^7)/7 + A(256*x^8)/8 + A(512*x^9)/9 + A(1024*x^10)/10 + ...).
%o (PARI) {a(n)=local(A=x); for(i=1, n, A=x*exp(sum(k=1, n, subst(A, x, 2^k*x^k +x*O(x^n))/k))); polcoeff(A, n)}
%o for(n=1,20,print1(a(n),", "))
%Y Cf. A000081, A229900, A229807.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Oct 03 2013
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