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A025242 Generalized Catalan numbers. 6
2, 1, 1, 2, 5, 13, 35, 97, 275, 794, 2327, 6905, 20705, 62642, 190987, 586219, 1810011, 5617914, 17518463, 54857506, 172431935, 543861219, 1720737981, 5459867166, 17369553427, 55391735455, 177040109419, 567019562429, 1819536774089 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Number of Dyck paths of semilength n-1 with no UUDD (n>1). Example: a(4)=2 because the only Dyck paths of semilength 3 with no UUDD in them are UDUDUD and UUDUDD (the nonqualifying ones being UDUUDD, UUDDUD and UUUDDD). - Emeric Deutsch, Jan 27 2003

a(n ) = number of Dyck (n-2)-paths with no DDUU (n>2). Example: a(6)=13 counts all 14 Dyck 4-paths except UUDDUUDD which contains a DDUU. There is a simple bijective proof: given a Dyck path that avoids DDUU, for every occurrence of UUDD except the first, the ascent containing this UU must be immediately preceded by a UD (else a DDUU would be present). Transfer the latter UD to the middle of the DD in the UUDD. Then insert a new UD in the middle of the first DD if any; if not, the path is a sawtooth UDUD...UD, in which case insert a UD at the end. This is a bijection from DDUU-avoiding Dyck n-paths to UUDD-avoiding Dyck (n+1)-paths. - David Callan, Sep 25 2006

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Nancy S. S. Gu, Nelson Y. Li, and Toufik Mansour, 2-Binary trees: bijections and related issues, Discr. Math., 308 (2008), 1209-1221.

Yvan Le Borgne, Counting Upper Interactions in Dyck Paths, Séminaire Lotharingien de Combinatoire, Vol. 54, B54f (2006), 16 pp.

V. Jelinek, T. Mansour, M. Shattuck, On multiple pattern avoiding set partitions, Adv. Appl. Math. 50 (2) (2013) 292-326, Theorem 4.1, without the leading 2.

T. Mansour, Restricted 1-3-2 permutations and generalized patterns, arXiv:math/0110039 [math.CO], 2001.

T. Mansour, Restricted 1-3-2 permutations and generalized patterns, Annals of Combin., 6 (2002), 65-76. (Example 2.10.)

T. Mansour and M. Shattuck, Restricted partitions and generalized Catalan numbers, PU. M. A., Vol. (2011), No. 2, pp. 239-251. - From N. J. A. Sloane, Oct 13 2012

L. Pudwell, Pattern-avoiding ascent sequences, Slides from a talk, 2015.

L. Pudwell and A. Baxter, Ascent sequences avoiding pairs of patterns, Slides, Permutation Patterns 2014, East Tennessee State University Jul 07 2014.

A. Sapounakis, I. Tasoulas and P. Tsikouras, Counting strings in Dyck paths, Discrete Math., 307 (2007), 2909-2924.

FORMULA

a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-3)*a(3) for n >= 4.

G.f.: (1+2*x+x^2-sqrt(1-4*x+2*x^2+x^4))/2 - Michael Somos, Jun 08, 2000.

Conjecture: n*(n+1)*a(n) +(n^2+n+2)*a(n-1) +2*(-9*n^2+15*n+17)*a(n-2) +2*(5*n+4)*(n-4)*a(n-3) +(n+1)*(n-6)*a(n-4) +(5*n+4)*(n-7)*a(n-5)=0. - R. J. Mathar, Jan 12 2013

G.f.: 2 + x - x*G(0), where G(k)= 1 - 1/(1 - x/(1 - x/(1 - x/(1 - x/(x - 1/G(k+1) ))))); (continued fraction). - Sergei N. Gladkovskii, Jul 12 2013

MATHEMATICA

a[ 0 ]=1; a[ n_Integer ] := a[ n ]=a[ n-1 ]+Sum[ a[ k ]*a[ n-1-k ], {k, 2, n-1} ];

PROG

(PARI) a(n)=polcoeff((1+2*x+x^2-sqrt(1-4*x+2*x^2+x^4+x*O(x^n)))/2, n)

CROSSREFS

Cf. A000108, A001006, A006318, A004148, A007477, A082582, A086581.

Sequence in context: A273488 A334955 A117848 * A163982 A246661 A246660

Adjacent sequences:  A025239 A025240 A025241 * A025243 A025244 A025245

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Olivier Gérard

STATUS

approved

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Last modified March 7 16:12 EST 2021. Contains 341897 sequences. (Running on oeis4.)