OFFSET
1,2
COMMENTS
a(n) is also the number of binary trees with n nodes such that each node and each edge is labeled uniquely from {1,...,2n-1} and the label of each node is less than the labels below it. - Brad R. Jones, Aug 13 2014
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..100
C. Coscia, J. DeWitt, F. Yang, Y. Zhang, Online and Random Domination of Graphs, arXiv preprint arXiv:1509.08876 [math.CO], 2015-2017.
B. R. Jones, On tree hook length formulas, Feynman rules and B-series, Master's thesis, Simon Fraser University, 2014.
D. E. Knuth, Whirlpool Permutations, May 05 2020
M. La Croix, A combinatorial proof of a result of Gessel and Greene, Discr. Math., 306 (2006), 2251-2256.
FORMULA
E.g.f. (odd powers only): Sum_{n>=1} a(n) * x^(2*n-1)/(2*n-1)! = tanh(x)/(1-x*tanh(x)).
a(n) = (2*n-1)!*Sum(m=1..2*n, Sum(k=0..2*n-2*m, binomial(k+m-1,m-1)*(k+m)!*(-1)^k*2^(2*n-2*m-k)*Stirling2(2*n-m,k+m))/(2*n-m)!). - Vladimir Kruchinin, Jun 14 2011
a(n) ~ 2*(2*n-1)! / r^(2*n+2), where r = 1.1996786402577... is the root of the equation (r-1)*exp(2*r) = r+1. - Vaclav Kotesovec, Dec 10 2013
MATHEMATICA
Table[n!*SeriesCoefficient[Tanh[x]/(1-x*Tanh[x]), {x, 0, n}], {n, 1, 41, 2}] (* Vaclav Kotesovec, Dec 10 2013 *)
PROG
(Maxima)
a(n):=(2*n-1)!*sum(sum(binomial(k+m-1, m-1)*(k+m)!*(-1)^k*2^(2*n-2*m-k)*stirling2(2*n-m, k+m), k, 0, 2*n-2*m)/(2*n-m)!, m, 1, 2*n); /* Vladimir Kruchinin, Jun 14 2011 */
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Sep 21 2006
STATUS
approved