

A025242


Generalized Catalan numbers.


7



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
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OFFSET

1,1


COMMENTS

Number of Dyck paths of semilength n1 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) is the number of Dyck (n2)paths with no DDUU (n>2). Example: a(6)=13 counts all 14 Dyck 4paths 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 DDUUavoiding Dyck npaths to UUDDavoiding Dyck (n+1)paths.  David Callan, Sep 25 2006
For n>1, a(n) is the number of cyclic permutations of [n1] that avoid the vincular pattern 1342, i.e., the pattern 1342 where the 1 and 3 must be adjacent. By the trivial Wilf equivalence, the same applies for 2431, 3124, and 4213.  Rupert Li, Jul 27 2021


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Nancy S. S. Gu, Nelson Y. Li, and Toufik Mansour, 2Binary trees: bijections and related issues, Discr. Math., 308 (2008), 12091221.
Yvan Le Borgne, Counting Upper Interactions in Dyck Paths, Séminaire Lotharingien de Combinatoire, Vol. 54, B54f (2006), 16 pp.
V. Jelinek, T. Mansour and M. Shattuck, On multiple pattern avoiding set partitions, Adv. Appl. Math. 50 (2) (2013) 292326, Theorem 4.1, without the leading 2.
Rupert Li, Vincular Pattern Avoidance on Cyclic Permutations, arXiv:2107.12353 [math.CO], 2021.
T. Mansour, Restricted 132 permutations and generalized patterns, arXiv:math/0110039 [math.CO], 2001.
T. Mansour, Restricted 132 permutations and generalized patterns, Annals of Combin., 6 (2002), 6576. (Example 2.10.)
T. Mansour and M. Shattuck, Restricted partitions and generalized Catalan numbers, PU. M. A., Vol. (2011), No. 2, pp. 239251.  From N. J. A. Sloane, Oct 13 2012
L. Pudwell, Patternavoiding 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), 29092924.


FORMULA

a(n) = a(1)*a(n1) + a(2)*a(n2) + ... + a(n3)*a(3) for n >= 4.
G.f.: (1+2*x+x^2sqrt(14*x+2*x^2+x^4))/2.  Michael Somos, Jun 08 2000
Conjecture: n*(n+1)*a(n) +(n^2+n+2)*a(n1) +2*(9*n^2+15*n+17)*a(n2) +2*(5*n+4)*(n4)*a(n3) +(n+1)*(n6)*a(n4) +(5*n+4)*(n7)*a(n5)=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[ n1 ]+Sum[ a[ k ]*a[ n1k ], {k, 2, n1} ];


PROG

(PARI) a(n)=polcoeff((1+2*x+x^2sqrt(14*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 * A356696 A163982 A246661
Adjacent sequences: A025239 A025240 A025241 * A025243 A025244 A025245


KEYWORD

nonn,easy


AUTHOR

Clark Kimberling, Olivier Gérard


STATUS

approved



