A006595 a(n) = (n+2)!/4 + n!/2.
1, 2, 7, 33, 192, 1320, 10440, 93240, 927360, 10160640, 121564800, 1576713600, 22034073600, 330032102400, 5274286617600, 89575694208000, 1611054821376000, 30589118816256000, 611426688897024000, 12833558093131776000, 282216632948490240000
Offset: 0
Examples
a(3) = 7. There are 3! = 6 non-plane recursive trees on 4 nodes shown below. The total number of nodes of outdegree 1 is 3+1+1+1+1+0 = 7. .0o......0o..........0o..........0o.........0o...........0o...... ..|.......|........../.\........./.\......../.\........../|\..... ..|.......|........./...\......./...\....../...\......../.|.\.... .1o......1o.......1o.....o3...1o....o2...2o.....o1...../..|..\... ..|....../.\.......|...........|..........|..........1o..2o...o3. ..|...../...\......|...........|..........|...................... .2o...2o.....o3...2o..........3o.........3o...................... ..|.............................................................. ..|.............................................................. .3o.............................................................. .................................................................
References
- L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 258.
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..200
- Dan Daly and Lara Pudwell, Pattern avoidance in rook monoids, Special Session on Patterns in Permutations and Words, Joint Mathematics Meetings, 2013. - From _N. J. A. Sloane_, Feb 03 2013
- Rui-Li Liu and Feng-Zhen Zhao, New Sufficient Conditions for Log-Balancedness, With Applications to Combinatorial Sequences, J. Int. Seq., Vol. 21 (2018), Article 18.5.7.
- J. R. Stembridge, Some combinatorial aspects of reduced words in finite Coxeter groups, Trans. Amer. Math. Soc. 349 (1997), no. 4, 1285-1332.
Crossrefs
A diagonal of A059418.
Programs
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Magma
[Factorial(n+2)/4+Factorial(n)/2: n in [0..25]]; // Vincenzo Librandi, Aug 26 2016
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Mathematica
Table[(n + 2)! / 4 + n! / 2, {n, 0, 30}] (* Vincenzo Librandi, Aug 26 2016 *) -
PARI
a(n) = (n+2)!/4 + n!/2; \\ Michel Marcus, Aug 04 2013
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SageMath
def A006595(n): return (n**2+3*n+4)*factorial(n)//4 print([A006595(n) for n in range(41)]) # G. C. Greubel, Aug 28 2025
Formula
E.g.f.: 1/2*(x^2-2*x+2)/(1-x)^3. - Peter Bala, Jul 08 2012
a(n) - (n+2)*a(n-1) + 2*a(n-2) - 2*(n-2)*a(n-3) = 0. - R. J. Mathar, May 30 2014
Extensions
Improved description and sequence extended by N. J. A. Sloane, Aug 15 1995
Comments