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%I #16 Sep 01 2024 18:22:55
%S 0,1,4,27,270,3645,62370,1295595,31689630,892387125,28439784450,
%T 1011998000475,39773696712750,1711186282730925,79990996596761250,
%U 4037168079574504875,218797477268743122750,12673229445076108033125
%N Number of topologically distinct trees with n vertices, including Steiner trees.
%C Let F(n,s) = number of Steiner trees with n vertices and s Steiner points; then A001147 is also F(n,n-2) for n>2. - _Robert G. Wilson v_, May 10 2005
%H E. N. Gilbert and H. O. Pollak, <a href="https://doi.org/10.1137/0116001">Steiner minimal trees</a>, SIAM J. Appl. Math. 16, (1968), pp. 1-29.
%F a(n) = Sum_{s=0..n-2} 2^(-s)*binomial(n, s+2)*(n+s-2)!/s!. - _Robert G. Wilson v_, May 10 2005
%F a(n) ~ 3^(3/4) * (2 + sqrt(3))^(n - 3/2) * n^(n-2) / exp(n). - _Vaclav Kotesovec_, Nov 27 2017
%e Let F(n,s) = number of Steiner trees with n vertices and s Steiner points. Then a(3)=4 because we can have F(3,0)=3 and F(3,1)=1.
%t f[n_] := Sum[ Binomial[n, k + 2](n + k - 2)!/(k!2^k), {k, 0, n - 2}]; Table[ f[n], {n, 18}] (* _Robert G. Wilson v_, May 10 2005 *)
%Y Cf. A001147.
%K easy,nonn
%O 1,3
%A Alexandre Goncalves (alexg(AT)civil.ist.utl.pt), Apr 22 2005
%E More terms from _Robert G. Wilson v_, May 10 2005