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A001761
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a(n) = (2*n)!/(n+1)!.
(Formerly M3635 N1478)
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15
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1, 1, 4, 30, 336, 5040, 95040, 2162160, 57657600, 1764322560, 60949324800, 2346549004800, 99638080819200, 4626053752320000, 233153109116928000, 12677700308232960000, 739781100339240960000, 46113021921146019840000
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Number of dissections of a disk.
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 (reinhard.zumkeller(AT)gmail.com), Feb 19 2010]
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REFERENCES
| 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.
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).
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LINKS
| 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, 4 (2001), #01.2.5.
K. A. Penson and A. I. Solomon, Coherent states from combinatorial sequences.
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FORMULA
| a(n+2) = sum(A038455(n, m), m=1..n), n >= 1 - Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)
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 (penson(AT)lptl.jussieu.fr), 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
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MAPLE
| seq(mul((n+k), k=2..n), n=0..17); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 15 2008
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PROG
| (Mupad) combinat::catalan(n)*n! $ n = 0..17; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 15 2007
(Other) 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]
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CROSSREFS
| Sequence in context: A128329 A006149 A121413 * A099712 A052316 A089918
Adjacent sequences: A001758 A001759 A001760 * A001762 A001763 A001764
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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