A104653 Number of topologically distinct trees with n vertices, including Steiner trees.

0, 1, 4, 27, 270, 3645, 62370, 1295595, 31689630, 892387125, 28439784450, 1011998000475, 39773696712750, 1711186282730925, 79990996596761250, 4037168079574504875, 218797477268743122750, 12673229445076108033125

Offset

1,3

Comments

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

References

Gilbert, E. N. and Pollak, H. O. (1968) Steiner minimal trees. SIAM Journal of Applied Mathematics, 16: 1-29.

Links

Table of n, a(n) for n=1..18.

Formula

a(1) = 0 a(2) = 1 a(n) = sum indexed on s(0 to n-2; 2^s * binomial(n, s+2) * (n+s-2)!/s!.

a(1) = 0, a(2) = 1; a(n) = sum_{s=0..n-2} (2^-s)* binomial(n, s+2)*(n+s-2)!/s!. - Robert G. Wilson v, May 10 2005

a(n) ~ 3^(3/4) * (2 + sqrt(3))^(n - 3/2) * n^(n-2) / exp(n). - Vaclav Kotesovec, Nov 27 2017

Example

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.

Mathematica

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 *)

Crossrefs

Cf. A001147.

Sequence in context: A265270 A161633 A052871 * A194787 A020558 A259485

Adjacent sequences: A104650 A104651 A104652 * A104654 A104655 A104656

Keyword

easy,nonn

Author

Alexandre Goncalves (alexg(AT)civil.ist.utl.pt), Apr 22 2005

Extensions

More terms from Robert G. Wilson v, May 10 2005

Status

approved