A000551 Number of labeled rooted trees of height 2 with n nodes.

6, 36, 200, 1170, 7392, 50568, 372528, 2936070, 24617120, 218521116, 2045278248, 20112821274, 207162957120, 2228888801040, 24989309310944, 291322555295886, 3524580202643136, 44176839081266340, 572725044269255640

Offset: 3

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

References

• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 3..200
• J. Riordan, Enumeration of trees by height and diameter, IBM J. Res. Dev. 4 (1960), 473-478.
• Index entries for sequences related to rooted trees
• Index entries for sequences related to trees

Programs

Maple

# ht(m) counts trees of height<=m; ht := proc(m) local i; [ T0,{seq(T.i=Prod(Z,Set(T.(i+1))),i=0..m-1),T.m=Z},labeled ] end: M[ 4220 ] := n->count(ht(2),size=n)-count(ht(1),size=n): seq(M[ 4220 ](n), n=3..21);

Mathematica

a[n_] = n*(n-1)*Sum[k^(n-2-k)*Binomial[n-2, k-1], {k, n-2}]; Table[a[n], {n, 3, 21}] (* Jean-François Alcover, Jun 29 2011, after formula *)
With[{nn=30},Drop[CoefficientList[Series[x (Exp[x Exp[x]]-Exp[x]),{x,0,nn}], x] Range[0,nn]!,3]] (* Harvey P. Dale, Apr 17 2017 *)

Formula

E.g.f.: x*(exp(x*exp(x))-exp(x)), cf. A074728. - Vladeta Jovovic, Jan 29 2008

a(n) = n*(n-1)*A074728(n-2). - Sean A. Irvine, Nov 21 2010

Extensions

More terms from Flajolet and Zimmermann, Mar 15 1996