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A064062 Generalized Catalan numbers C(2; n). 22
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)
OFFSET

0,3

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

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.

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.

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

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]

PROG

(PARI) a(n)=polcoeff((3-sqrt(1-8*x+x*O(x^n)))/(2+2*x), n)

CROSSREFS

Sequence in context: A027277 A200754 A062992 * A114191 A107592 A028418

Adjacent sequences:  A064059 A064060 A064061 * A064063 A064064 A064065

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Sep 13 2001

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Last modified February 17 06:27 EST 2012. Contains 205998 sequences.