A108521 - OEIS

%I #25 Aug 06 2020 18:25:03

%S 1,2,5,16,53,194,730,2868,11526,47370,197786,837467,3585696,15501423,

%T 67563442,296579626,1309973823,5817855174,25964218471,116379947718,

%U 523699384013,2364967753113,10714396241046,48684193997623

%N Number of rooted trees with n generators.

%C A generator is a leaf or a node with just one child.

%H Andrew Howroyd, <a href="/A108521/b108521.txt">Table of n, a(n) for n = 1..200</a>

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EulerTransform.html">Euler Transform</a>.

%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>

%F G.f.: satisfies (2-x)*A(x) = x - 1 + EULER(A(x)).

%t 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 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 (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}

%o 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 Cf. A000081, A000669, A007151, A108522 - A108529, A335342 (free trees).

%K nonn

%O 1,2

%A _Christian G. Bower_, Jun 07 2005