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

0,3

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

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

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 *)

Prog

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

Crossrefs

Cf. A001620.

Sequence in context: A011967 A250886 A055732 * A039625 A020020 A000882

Adjacent sequences: A125059 A125060 A125061 * A125063 A125064 A125065

Keyword

nonn,changed

Author

Wenjin Woan, Jan 09 2007

Extensions

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

Status

approved