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%I A319369 #19 Mar 16 2024 17:23:32
%S A319369 1,3,28,430,9376,269675,9632960,411395268,20445999734,1159248404721,
%T A319369 73846864163348,5221802726902476,405858598184643930,
%U A319369 34392275731729465799,3155760058245300968416,311720334688779807141832,32980137195294216968253900,3720954854814866649904474180
%N A319369 Number of series-reduced rooted trees with n leaves of n colors.
%C A319369 Not all of the n colors need to be used.
%H A319369 Alois P. Heinz, <a href="/A319369/b319369.txt">Table of n, a(n) for n = 1..340</a>
%H A319369 V. P. Johnson, <a href="http://people.math.sc.edu/czabarka/Theses/JohnsonThesis.pdf">Enumeration Results on Leaf Labeled Trees</a>, Ph. D. Dissertation, Univ. Southern Calif., 2012.
%F A319369 a(n) ~ c * d^n * n^(n - 3/2), where d = 1/(2*log(2) - 1) = 2.588699449562089830805384431942090... and c = 0.2580000331300831455241033648... - _Vaclav Kotesovec_, Sep 18 2019, updated Mar 16 2024
%p A319369 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A319369       add(binomial(A(i, k)+j-1, j)*b(n-i*j, i-1, k), j=0..n/i)))
%p A319369     end:
%p A319369 A:= (n, k)-> `if`(n<2, n*k, b(n, n-1, k)):
%p A319369 a:= n-> A(n$2):
%p A319369 seq(a(n), n=1..20);  # _Alois P. Heinz_, Sep 18 2018
%t A319369 b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[A[i, k] + j - 1, j]*b[n - i*j, i - 1, k], {j, 0, n/i}]]];
%t A319369 A[n_, k_] := If[n < 2, n*k, b[n, n - 1, k]];
%t A319369 a[n_] := A[n, n];
%t A319369 a /@ Range[1, 20] (* _Jean-François Alcover_, Sep 24 2019, after _Alois P. Heinz_ *)
%o A319369 (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
%o A319369 a(n)={my(v=[n]); for(n=2, n, v=concat(v, EulerT(concat(v, [0]))[n])); v[n]}
%Y A319369 Main diagonal of A319254.
%Y A319369 Cf. A000311 (1 leaf of each color), A316651.
%K A319369 nonn
%O A319369 1,2
%A A319369 _Andrew Howroyd_, Sep 17 2018

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