A086677 Number of Steiner topologies on n points.

1, 4, 31, 360, 5625, 110880, 2643795, 74035080, 2382538725, 86656878000, 3515761193175, 157425426358200, 7711961781949425, 410298436511964000, 23559634669682986875, 1452240056377167057000, 95649328231839993736125

Offset: 2

Author

N. J. A. Sloane, Jul 28 2003

Keywords

• nonn
• easy
• nice

References

• F. K. Hwang, D. S. Richards and P. Winter, The Steiner Tree Problem, North-Holland, 1992, see p. 14.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 2..100

Crossrefs

Cf. A001147.

Programs

Mathematica

a[n_] := Sum[ Binomial[n, i]*Binomial[n-i, i+k+2]*(n+k-2)!*(2i+2k-1)!!*(i+k)! / ((2*(i+k))!*k!), {k, 0, n-2}, {i, 0, (n-k-2)/2}]; Table[a[n], {n, 2, 18}] (* Jean-François Alcover, Sep 03 2012, after formula *)

Formula

Let f(n) = (2*n-4)!/(2^(n-2)*(n-2)!) (A001147) and let F(n, k) = binomial(n, k+2) f(k) (n+k-2)! / (2k)!. Then a(n) = Sum_{k=0..n-2} Sum_{i=0..floor((n-k-2)/2)} binomial(n, i) F(n-i, k+i) (k+i)! / k!.

E.g.f. (for offset 0): 4*(x-3)/(x+1)^4 - (-13+22*x+3*x^2)/((-x^2-4*x+1)^(1/2)*(x+1)^4). - Mark van Hoeij, Oct 31 2011

a(n) ~ 1/8*sqrt(250-110*sqrt(5)) * n^(n-2) * (2+sqrt(5))^n / exp(n). - Vaclav Kotesovec, Mar 10 2014

Equivalently, a(n) ~ 5^(3/4) * phi^(3*n - 5/2) * n^(n-2) / (4 * exp(n)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 07 2021

Extensions

More terms from Vladeta Jovovic, Jul 29 2003