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A125062
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Number of increasing trees with hills of height 1.
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1
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1, 1, 4, 15, 68, 370, 2364, 17388, 144864, 1349136, 13894560, 156831840, 1925527680, 25550778240, 364416917760, 5559659078400, 90349397913600, 1558170228787200, 28423674336153600, 546807873520742400, 11064204944529408000, 234902850943703040000, 5221386564941352960000
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OFFSET
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0,3
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COMMENTS
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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
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REFERENCES
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R. P. Stanley, Enumerative Combinatorics, Vol. 1, Cambridge University Press, 1997, p25.
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LINKS
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FORMULA
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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
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MAPLE
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a := n -> ifelse(n = 0, 1, (n - 1)! * (n*(harmonic(n) + 1) - 1)):
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MATHEMATICA
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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 *)
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PROG
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(PARI) x='x+O('x^30); Vec(serlaplace((1+x*log(1/(1-x)))/(1-x))) \\ G. C. Greubel, Aug 31 2018
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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Edited by the Associate Editors of the OEIS, Oct 05 2009
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STATUS
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approved
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