0,4

Also number of Dyck paths of semilength n with no UDUD. Example: a(3)=3 because the only Dyck paths of semilength 3 with no UDUD in them are: UDUUDD, UUDDUD and UUUDDD (the nonqualifying ones being UUDUDD and UDUDUD). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 27 2003

Contribution from Paul Barry (pbarry(AT)wit.ie), Jan 29 2009: (Start)

The sequence 1,1,1,3,7,19,... has general term sum{k=0..n, C(n+k,2k)*(-1)^(n-k)*A006318(k)} and g.f. given by

1/(1+x-2x/(1+x-x/(1+x-2x/(1+x-x/(1+x-2x/(1+x-x/(1-..... (continued fraction). (End)

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

M. D. Atkinson and T. Stitt, Restricted permutations and the wreath product, Discrete Math., 259 (2002), 19-36.

Toufik Mansour, Statistics on Dyck Paths, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.5.

G.f.: (1-x-x^2-(1-2*x-5*x^2-2*x^3+x^4)^(1/2))/(2*x+2*x^2).

G.f. A(x) satisfies A(x)=x+(x+x^2)(A(x)+A(x)^2). - Michael Somos Sep 08 2005

Given g.f. A(x), then series reversion of B(x)=x+x*A(x) is -B(-x). - Michael Somos Sep 08 2005

Given g.f. A(x), then B(x)=x+x*A(x) satisfies 0=f(x, B(x)) where f(u, v)=u^2*(v+v^2)+u*(1+v+v^2)-v. - Michael Somos Sep 08 2005

Given g.f. A(x), then B(x)=x+x*A(x) satisfies B(x)=x+C(x*B(x)) where C(x) is g.f. of A006318 with offset 1. - Michael Somos Sep 08 2005

(PARI) {a(n)=if(n<1, 0, polcoeff( 2*(1+x)/(1+x+x^2 +sqrt((1-x+x^2)^2 -8*x^2+x*O(x^n))), n))} /* Michael Somos Sep 08 2005 */

Sequence in context: A059506 A007575 A026299 this_sequence A104522 A115760 A100702

Adjacent sequences: A078478 A078479 A078480 this_sequence A078482 A078483 A078484

nonn

N. J. A. Sloane (njas(AT)research.att.com), Jan 04 2003