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

1,3

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.

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

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 *)

Prog

(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

Crossrefs

Sequence in context: A111726 A090376 A232042 * A073479 A147690 A350830

Adjacent sequences: A125304 A125305 A125306 * A125308 A125309 A125310

Keyword

nonn

Author

Wenjin Woan, Jan 17 2007

Extensions

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

Status

approved