0,4

Words of length n in language defined by L = 1 + a + (L)L: L(0)=1, L(1)=a, L(2)=(), L(3)=(a)+()a, L(4)=(())+(a)a+()(), ...

G.f. A(x) satisfies the equation 0=1+x-A(x)+(xA(x))^2.

Series reversion of xA(x) is x*A082582(-x). - Michael Somos, Jul 22 2003

a(n) = number of Motzkin n-paths (A001006) in which no flatstep (F) is immediately followed by either an upstep (U) or a flatstep, in other words, each flatstep is either followed by a downstep (D) or ends the path. For example, a(4)=3 counts UDUD, UFDF, UUDD. - David Callan (callan(AT)stat.wisc.edu), Jun 07 2006

a(n) = number of Dyck (n+1)-paths (A000108) containing no UDUs and no subpaths of the form UUPDD where P is a nonempty Dyck path. For example, a(4)=3 counts UUUDDUUDDD, UUDDUUDDUD, UUUDDUDDUD. - David Callan (callan(AT)stat.wisc.edu), Sep 25 2006

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

M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 441

a(n)=sum(a(k)a(n-2-k)), n>1.

The g.f. satisfies A(x)-x^2A(x)^2 = 1+x. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Jun 30 2003

G.f.: (1-sqrt(1-4x^2-4x^3))/(2x^2).

G.f.: (1+x)c(x^2(1+x)) where c(x) is g.f. of A000108. - Paul Barry (pbarry(AT)wit.ie), May 31 2006

A007477 := proc(n) option remember; local k; if n <= 1 then 1 else add(A007477(k)*A007477(n-k-2), k=0..n-2); fi; end;

(PARI) a(n)=polcoeff((1-sqrt(1-4*x^2-4*x^3+x^3*O(x^n)))/2, n+2)

Cf. A115178.

Adjacent sequences: A007474 A007475 A007476 this_sequence A007478 A007479 A007480

Sequence in context: A063895 A027214 A132831 this_sequence A096202 A036653 A123769

nonn,nice,easy

njas

Additional comments from Michael Somos, Aug 03 2000.