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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
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OFFSET
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0,3
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COMMENTS
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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, Mar 30 2007
The Hankel transform of this sequence is A002416 . - Philippe DELEHAM, 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, Apr 14 2008
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REFERENCES
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Paul Barry and Aoife Hennessy, Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8.- From N. J. A. Sloane, Oct 08 2012
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
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Table of n, a(n) for n=0..21.
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
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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) }. - Philippe Deléham, 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, 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
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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, Aug 27 2009]
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PROG
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(PARI) a(n)=polcoeff((3-sqrt(1-8*x+x*O(x^n)))/(2+2*x), n)
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CROSSREFS
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Sequence in context: A027277 A200754 A062992 * A114191 A107592 A215257
Adjacent sequences: A064059 A064060 A064061 * A064063 A064064 A064065
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KEYWORD
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nonn,easy
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AUTHOR
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Wolfdieter Lang, Sep 13 2001
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STATUS
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approved
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