0,3

Series reversion of g.f. A(x) is -A(-x). - Michael Somos, Jul 27 2003

a(n) = number of royal paths (A006318) from (0,0) to (n-1,n-1) such that every northeast (diagonal) step is either immediately followed by a north step or ends the path. For example a(3)=4 counts EDN, EENN, END, ENEN (E=east,D=diagonal,N=north). - David Callan (callan(AT)stat.wisc.edu), Jul 03 2006

Comments from David Callan (callan(AT)stat.wisc.edu), Sep 25 2006: a(n) = # ordered trees with n leaves in which (i) every node (= non-root non-leaf vertex) has at least 2 children, and (ii) each leaf is either the leftmost or rightmost child of its parent. For example, a(3)=4 counts

......|

./\.../ \

./\..../\

and their mirror images.

D. Merlini, D. G. Rogers, R. Sprugnoli and M. C. Verri, On some alternative characterizations of Riordan arrays, Canad. J. Math., 49 (1997), 301-320.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 655

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 657

D. Merlini, D. G. Rogers, R. Sprugnoli and M. C. Verri, On some alternative characterizations of Riordan arrays, Canad. J. Math., 49 (1997), 301-320.

a(n)=sum(C(n-k-1)*binomial(n-k, k), k=0..floor(n/2)), where C(q)=binomial(2q, q)/(q+1) are the Catalan numbers (A000108). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 14 2001

na(n)=(4n-6)a(n-1)+(4n-12)a(n-2), n>2. a(1)=1, a(2)=2.

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

a(n)=sum{k=0..n-1, C(k)C(k+1, n-k-1)} - Paul Barry (pbarry(AT)wit.ie), Feb 23 2005

G.f. A(x) satisfies A(x)=x+C(2x*A(x)) where C(x) is g.f. of Catalan numbers A000108 offset 1. - Michael Somos Sep 08 2005

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

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

a(n)=A052709(n)+A052709(n-1).

A100238(n)=-(-1)^n*a(n), if n>1.

Sequence in context: A113179 A056236 A028329 this_sequence A119430 A074034 A062962

Adjacent sequences: A025224 A025225 A025226 this_sequence A025228 A025229 A025230

nonn

Clark Kimberling (ck6(AT)evansville.edu)