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A108521 Number of rooted trees with n generators.

%I A108521 #25 Aug 06 2020 18:25:03
%S A108521 1,2,5,16,53,194,730,2868,11526,47370,197786,837467,3585696,15501423,
%T A108521 67563442,296579626,1309973823,5817855174,25964218471,116379947718,
%U A108521 523699384013,2364967753113,10714396241046,48684193997623
%N A108521 Number of rooted trees with n generators.
%C A108521 A generator is a leaf or a node with just one child.
%H A108521 Andrew Howroyd, <a href="/A108521/b108521.txt">Table of n, a(n) for n = 1..200</a>
%H A108521 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A108521 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EulerTransform.html">Euler Transform</a>.
%H A108521 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F A108521 G.f.: satisfies (2-x)*A(x) = x - 1 + EULER(A(x)).
%t A108521 a[1] = 1; a[n_] := a[n] = 1+a[n-1]+Total[Product[Binomial[a[i]-1+Count[#,i], Count[#,i]], {i, DeleteCases[DeleteDuplicates[#],1]}]&/@ IntegerPartitions[n,{2,n-1}]]; Table[a[n],{n,24}] (* _Robert A. Russell_, Jun 02 2020 *)
%t A108521 a[1] = 1; a[n_] := a[n] = a[n-1] + (DivisorSum[n, a[#] # &, #<n &] + Sum[c[k] b[n-k], {k,1,n-1}])/n; b[n_] := b[n] = (c[n] + Sum[c[k] b[n-k], {k,1,n-1}])/n; c[n_] := c[n] = DivisorSum[n, a[#] # &]; Table[a[k], {k, 24}] (* _Robert A. Russell_, Jun 04 2020 *)
%o A108521 (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
%o A108521 seq(n)={my(v=[1]); for(n=2, n, v=concat(v, v[#v] + EulerT(concat(v,[0]))[n])); v} \\ _Andrew Howroyd_, Aug 31 2018
%Y A108521 Cf. A000081, A000669, A007151, A108522 - A108529, A335342 (free trees).
%K A108521 nonn
%O A108521 1,2
%A A108521 _Christian G. Bower_, Jun 07 2005