A125062 - OEIS

%I #31 Apr 09 2024 17:03:11

%S 1,1,4,15,68,370,2364,17388,144864,1349136,13894560,156831840,

%T 1925527680,25550778240,364416917760,5559659078400,90349397913600,

%U 1558170228787200,28423674336153600,546807873520742400,11064204944529408000,234902850943703040000,5221386564941352960000

%N Number of increasing trees with hills of height 1.

%C 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

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

%H Vincenzo Librandi, <a href="/A125062/b125062.txt">Table of n, a(n) for n = 0..200</a>

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

%F 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

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

%F a(0) = 1; For n > 0; a(n) = (n - 1)*(n - 1)! + abs(Stirling1(n + 1, 2)). - _Detlef Meya_, Apr 09 2024

%p a := n -> ifelse(n = 0, 1, (n - 1)! * (n*(harmonic(n) + 1) - 1)):

%p seq(a(n), n = 0..22);  # _Peter Luschny_, Apr 09 2024

%t 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 *)

%t 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 *)

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

%Y Cf. A001620.

%K nonn

%O 0,3

%A _Wenjin Woan_, Jan 09 2007

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