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A064062
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Generalized Catalan numbers C(2; n).
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22
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1, 1, 3, 13, 67, 381, 2307, 14589, 95235, 636925, 4341763, 30056445, 210731011, 1493303293, 10678370307, 76957679613, 558403682307, 4075996839933, 29909606989827, 220510631755773, 1632599134961667, 12133359132082173
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| a(n+1)= Y_{n}(n+1)= Z_{n}, n >= 0, in the Derrida et al. 1992 reference (see A064094) for alpha=2, beta =1 (or alpha=1, beta=2).
a(n) = number of Dyck n-paths (A000108) in which each upstep (U) not at ground level is colored red (R) or blue (B). For example, a(3)=3 counts URDD, UBDD, UDUD (D=downstep). - David Callan (callan(AT)stat.wisc.edu), Mar 30 2007
The Hankel transform of this sequence is A002416 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 19 2007
The sequence a(n)/2^n, with g.f. 1/(1-xc(x)/2), has Hankel transform 1/2^n. - Paul Barry (pbarry(AT)wit.ie), Apr 14 2008
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REFERENCES
| N. Bonichon, C. Gavoille And N. Hanusse. Canonical Decomposition of Outerplanar Maps and Application to Enumeration, Coding and Generation. In Proceedings of WG'03, volume 2880 of LNCS, pp. 81-92, 2003.
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LINKS
| Alexander Burstein, Sergi Elizalde and Toufik Mansour, Restricted Dumont permutations, Dyck paths and noncrossing partitions, arXiv math.CO/0610234.
A. Vieru, Agoh's conjecture: its proof, its generalizations, its analogues, Arxiv preprint arXiv:1107.2938, 2011.
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FORMULA
| G.f.: (1+2*x*c(2*x))/(1+x) = 1/(1-x*c(2*x)) with c(x) g.f. of Catalan numbers A000108.
a(n)= A062992(n-1) = sum((n-m)*binomial(n-1+m, m)*(2^m)/n, m=0..n-1), n >= 1, a(0) := 1.
a(n) = Sum{ k= 0...n, A059365(n, k)*2^(n-k) }. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jan 19 2004
G.f.: 1/(1-x/(1-2x/(1-2x/(1-2x/(1-.... =1/(1-x-2x^2/(1-4x-4x^2/(1-4x-4x^2/(1-.... (continued fractions). [From Paul Barry (pbarry(AT)wit.ie), Jan 30 2009]
a(n) = (32/Pi)*integral_{x=0..1} (8*x)^(n-1)*sqrt(x*(1-x)) / (8*x+1). [Groux Roland, Dec 12 2010]
a(n+2) = 8^(n+2)*( c(n+2)-c(1)*c(n+1)-sum_{i=0..n-1} 8^(-i-2)*c(n-i)*a(i+2) ) with c(n)=Catalan(n+2)/2^(2*n+1). [Groux Roland, Dec 12 2010]
a(n) = the upper left term in M^n, M = the production matrix:
1, 1
2, 2, 1
4, 4, 2, 1
8, 8, 4, 2, 1
...
- Gary W. Adamson, July 08 2011
Conjecture: n*a(n) +(12-7n)*a(n-1)+4*(3-2n)*a(n-2)=0. - R. J. Mathar, Nov 16 2011
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MATHEMATICA
| a[0]=1; a[1]=1; a[n_]/; n>=2 := a[n] = a[n-1] + Sum[(a[k] + a[k-1])a[n-k], {k, n-1}]; Table[a[n], {n, 0, 10}] [From David Callan (callan(AT)stat.wisc.edu), Aug 27 2009]
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PROG
| (PARI) a(n)=polcoeff((3-sqrt(1-8*x+x*O(x^n)))/(2+2*x), n)
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CROSSREFS
| Sequence in context: A027277 A200754 A062992 * A114191 A107592 A028418
Adjacent sequences: A064059 A064060 A064061 * A064063 A064064 A064065
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KEYWORD
| nonn,easy
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Sep 13 2001
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