A128614 Number of labeled plane trees with n edges in which no vertex has outdegree one.

0, 6, 24, 360, 4320, 75600, 1451520, 33022080, 841881600, 24069830400, 759217536000, 26234438630400, 984940333977600, 39932452175616000, 1738621071323136000, 80909997706137600000, 4007834720796082176000

Offset

1,2

Links

G. C. Greubel, Table of n, a(n) for n = 1..375

W. Y. C. Chen, E. Y. P. Deng and L. L. M. Yang, Riordan paths and derangements, arXiv:math/0602298 [math.CO], 2006.

Formula

a(n) = n!*Sum_{k=1..n-1} binomial(n+1, k)*binomial(n-k-1, k-1).

a(n) = (n+1)!*A005043(n).

Mathematica

a[n_] := (-1)^n (n+1)! Hypergeometric2F1[-n, 1/2, 2, 4]; Array[a, 17] (* Jean-François Alcover, Feb 17 2019 *)

Prog

(Magma)
A128614:= func< n | n eq 1 select 0 else Factorial(n)*(&+[Binomial(n+1, k)*Binomial(n-k-1, k-1): k in [1..n-1]]) >;
[A128614(n): n in [1..40]]; // G. C. Greubel, Mar 26 2024
(SageMath)
def A128614(n): return factorial(n)*sum(binomial(n+1, k)*binomial(n-k-1, k-1) for k in range(1, n))
[A128614(n) for n in range(1, 41)] # G. C. Greubel, Mar 26 2024

Crossrefs

Cf. A005043.

Sequence in context: A010567 A097171 A152886 * A285018 A139240 A052524

Adjacent sequences: A128611 A128612 A128613 * A128615 A128616 A128617

Keyword

nonn

Author

Ralf Stephan, May 08 2007

Status

approved