A001761 a(n) = (2*n)!/(n+1)!.

1, 1, 4, 30, 336, 5040, 95040, 2162160, 57657600, 1764322560, 60949324800, 2346549004800, 99638080819200, 4626053752320000, 233153109116928000, 12677700308232960000, 739781100339240960000, 46113021921146019840000

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

According to the Beineke and Pippert paper, the number of dissections of a disk is given by D(n)=R(n)/(n-2)!, where R(n)=A001761(n-2) is the number of labeled planar 2-trees having n vertices and rooted at a given exterior edge. [Clarified by M. F. Hasler, Feb 22 2012]

a(n+1) is the number of labeled incomplete ternary trees on n vertices in which each left and middle child have a larger label than their parent. - Brian Drake, Jul 28 2008

For n>0: a(n) = A173333(2*n,n+1); cf. A006963, A001813. - Reinhard Zumkeller, Feb 19 2010

References

• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• 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
• L. W. Beineke and R. E. Pippert, Enumerating labeled k-dimensional trees and ball dissections, pp. 12-26 of Proceedings of Second Chapel Hill Conference on Combinatorial Mathematics and Its Applications, University of North Carolina, Chapel Hill, 1970. Reprinted in Math. Annalen, Vol. 191 (1971), pp. 87-98.
• Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
• Peter J. Cameron, Some treelike objects, Quart. J. Math. Oxford, Vol. 38, No. 2 (1987), pp. 155-183. See p. 166. - _N. J. A. Sloane_, Apr 18 2014
• Ali Chouria, Vlad-Florin Drǎgoi, and Jean-Gabriel Luque, On recursively defined combinatorial classes and labelled trees, arXiv:2004.04203 [math.CO], 2020.
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 80.
• K. A. Penson and J.-M. Sixdeniers, Integral Representations of Catalan and Related Numbers, J. Integer Sequences, Vol. 4 (2001), Article 01.2.5.
• Karol A. Penson and Allan I. Solomon, Coherent states from combinatorial sequences, arXiv:quant-ph/0111151, 2001.

Crossrefs

Cf. A000108, A173333, A006963, A001813.

Main diagonal of A255982, A256061.

Programs

Maple

seq(mul((n+k), k=2..n), n=0..17); # Zerinvary Lajos, Feb 15 2008

Mathematica

Table[(2*n)!/(n+1)!,{n,0,20}] (* Vincenzo Librandi, Feb 23 2012 *)

MuPAD

combinat::catalan(n)*n! $ n = 0..17; // Zerinvary Lajos, Feb 15 2007

PARI

A001761(n)=binomial(2*n,n+1)*(n-1)!  \\ M. F. Hasler, Feb 23 2012

PARI

{Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}
{a(n)=sum(k=0,n,(-1)^(n-k)*(n+1)^(k-1)*Stirling1(n,k))} \\ Paul D. Hanna, Nov 09 2012

Sage

[binomial(2*n,n)/(1+n)*factorial(n) for n in range(0, 18)] # Zerinvary Lajos, Dec 03 2009

Formula

a(n) = n!*Catalan(n) =n!* A000108(n). - N. J. A. Sloane, Apr 18 2014

a(n+2) = sum(A038455(n, m), m=1..n), n >= 1. - Wolfdieter Lang

E.g.f. for this sequence = o.g.f. for A000108. - Len Smiley, Dec 07 2001

Integral representation as the moment of a positive function on the positive half-axis: in Maple notation, a(n)=int(x^n*(-1/2+exp(-x/4)/sqrt(Pi*x)+erf(sqrt(x)/2)/2), x=0..infinity), n=0, 1... This representation is unique. - Karol A. Penson, Aug 21 2001

G.f.: If G_N(x)=1+sum('(2*k)!*(x^k)/(k+1)!', 'k'=1..N), G_N(x)=1+2*x/(G(0)-2*x); G(k)=4*x*(k^2)+6*k*x+k+2*x+2-2*x*(2*k+3)*((k+2)^2)/G(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Nov 24 2011

a(n) = Sum_{k=0..n} (-1)^(n-k) * (n+1)^(k-1) * Stirling1(n,k). - Paul D. Hanna, Nov 09 2012

G.f.: Q(0) where Q(k) = 1 + x*(2*k+1)*(4*k+1)/(k+1 - 4*x*(k+1)^2*(4*k+3)/(4*x*(k+1)*(4*k+3) + (2*k+3)/Q(k+1) )); (continued fraction ). - Sergei N. Gladkovskii, Apr 05 2013

G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x/(x + (k+2)/(2*k+2)/(2*k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 03 2013

Let A(x) = sum(k>=0, a(k)*x^k /(2*k)! ) = ( exp(x)-1)/x, then A(x) = 1/Q(0), where Q(k) = 1 - x/( 1 + (2*k+1)/(1 - x/( 1 + 2*(k+1)/Q(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Nov 24 2013

From Ilya Gutkovskiy, Jan 21 2017: (Start)

a(n) ~ sqrt(2)*4^n*n^(n-1)/exp(n).

Sum_{n>=0} 1/a(n) = (7*exp(1/4)*sqrt(Pi)*erf(1/2) + 10)/8 = 2.2865189388213215..., where erf() is the error function. (End)

D-finite with recurrence: (n+1)*a(n) -2*n*(2*n-1)*a(n-1)=0. - R. J. Mathar, Feb 16 2020

Sum_{n>=0} (-1)^n/a(n) = 3/4 - 5*sqrt(Pi)*erfi(1/2)/(8*exp(1/4)), where erfi() is the imaginary error function. - Amiram Eldar, Apr 03 2022