|
| |
|
|
A001761
|
|
a(n) = (2*n)!/(n+1)!.
(Formerly M3635 N1478)
|
|
17
|
|
|
|
1, 1, 4, 30, 336, 5040, 95040, 2162160, 57657600, 1764322560, 60949324800, 2346549004800, 99638080819200, 4626053752320000, 233153109116928000, 12677700308232960000, 739781100339240960000, 46113021921146019840000
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,3
|
|
|
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 (bdrake(AT)brandeis.edu), Jul 28 2008
For n>0: a(n) = A173333(2*n,n+1); cf. A006963, A001813. [From 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
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 80
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, 191 (1971), 87-98.
K. A. Penson and J.-M. Sixdeniers, Integral Representations of Catalan and Related Numbers, J. Integer Sequences, 4 (2001), #01.2.5.
K. A. Penson and A. I. Solomon, Coherent states from combinatorial sequences, arXiv:quant-ph/0111151.
|
|
|
FORMULA
|
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 (smiley(AT)math.uaa.alaska.edu), 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
n!*binomial(2*n,n)/(n+1) or A000108*n! - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 06 2006
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
|
|
|
MAPLE
|
seq(mul((n+k), k=2..n), n=0..17); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 15 2008
|
|
|
MATHEMATICA
|
Table[(2*n)!/(n+1)!, {n, 0, 20}] (* Vincenzo Librandi, Feb 23 2012 *)
|
|
|
PROG
|
(Mupad) combinat::catalan(n)*n! $ n = 0..17; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 15 2007
(Sage) [binomial(2*n, n)/(1+n)*factorial(n) for n in xrange(0, 18)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 03 2009]
(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
|
|
|
CROSSREFS
|
Sequence in context: A006149 A207833 A121413 * A099712 A209440 A052316
Adjacent sequences: A001758 A001759 A001760 * A001762 A001763 A001764
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
N. J. A. Sloane.
|
|
|
STATUS
|
approved
|
| |
|
|
