A125062 Number of increasing trees with hills of height 1.

1, 1, 4, 15, 68, 370, 2364, 17388, 144864, 1349136, 13894560, 156831840, 1925527680, 25550778240, 364416917760, 5559659078400, 90349397913600, 1558170228787200, 28423674336153600, 546807873520742400, 11064204944529408000, 234902850943703040000, 5221386564941352960000

Offset: 0

Author

Wenjin Woan, Jan 09 2007

Keywords

• nonn

Comments

If we discard the first 1 and set a(0)=1,a(1)=4, then a(n) = (n+1)!(H(n)+1), where H(n) = Sum_{k=1..n} 1/k. - Gary Detlefs, Jul 21 2010

References

• R. P. Stanley, Enumerative Combinatorics, Vol. 1, Cambridge University Press, 1997, p25.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A001620.

Programs

Maple

a := n -> ifelse(n = 0, 1, (n - 1)! * (n*(harmonic(n) + 1) - 1)):
seq(a(n), n = 0..22);  # Peter Luschny, Apr 09 2024

Mathematica

With[{nn=20},CoefficientList[Series[(1+x Log[1/(1-x)])/(1-x),{x,0,nn}], x]Range[0,nn]!] (* Harvey P. Dale, Mar 14 2012 *)
a[0]=1;a[n_]:=(n-1)*(n-1)!+Abs[StirlingS1[n+1,2]];Flatten[Table[a[n],{n,0,19}]] (* Detlef Meya, Apr 09 2024 *)

PARI

x='x+O('x^30); Vec(serlaplace((1+x*log(1/(1-x)))/(1-x))) \\ G. C. Greubel, Aug 31 2018

Formula

E.g.f.: (1+x*log(1/(1-x)))/(1-x).

a(n) = 2*(n-1)*a(n-1) - (n^2-4*n+5)*a(n-2) - (n-3)*(n-2)*a(n-3). - Vaclav Kotesovec, Nov 19 2012

a(n) ~ n!*(log(n) + gamma + 1 + O(log(n)/n)), where gamma is Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Nov 19 2012

a(0) = 1; For n > 0; a(n) = (n - 1)*(n - 1)! + abs(Stirling1(n + 1, 2)). - Detlef Meya, Apr 09 2024

Extensions

Edited by the Associate Editors of the OEIS, Oct 05 2009