A125307 Number of increasing trees with branches of height 1.

1, 1, 4, 15, 80, 480, 3444, 27790, 253504, 2556792, 28382880, 343071168, 4490999424, 63253633872, 954133373088, 15343385194800, 262060291958784, 4737396899952384, 90370907329842432, 1814141041750834560, 38229440785429201920, 843786230514306621696

Offset: 1

Author

Wenjin Woan, Jan 17 2007

Keywords

• nonn

References

• R. P. Stanley, Enumerative Combinatorics, Vol. 1, Cambridge University Press, 1997. Proposition 1.3.16, p. 25.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..448
• Vladimir Kruchinin, D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.

Programs

Mathematica

Range[0, 21]!CoefficientList[ Series[(x - 1 + Log[1 - x])/((1 - x)^2(Log[1 - x] - 1)), {x, 0, 21}], x] (* Robert G. Wilson v, Jan 26 2007 *)

Maxima

a(n):=n!*(sum((-1)^(m)*(n-m+1)/(m-1)!*sum(k!*stirling1(m-1,k), k,1,m-1), m,2,n)+1); /* Vladimir Kruchinin, Sep 09 2010 */

PARI

x='x+O('x^30); Vec(serlaplace( (x-1+log(1-x))/((x-1)^2*(log(1-x) -1)))) \\ G. C. Greubel, Sep 05 2018

Formula

E.g.f.: (x-1+log(1-x)) / ( (x-1)^2 (log(1-x)-1) ).

a(n) = n!*(sum((-1)^(m)*(n-m+1)/(m-1)!*sum(k!*Stirling1(m-1,k),k,1,m-1),m,2,n)+1). - Vladimir Kruchinin, Sep 09 2010

a(n) ~ n!*n*(1 - 1/log(n) + gamma/log(n)^2), where gamma is Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Sep 25 2013

Extensions

More terms from N. J. A. Sloane, Jan 26 2007