0,2

More generally coefficients of (1-m*x-sqrt(m^2*x^2-(2*m+4)*x+1))/(2*x) are given by a(0)=1 and n>0 a(n)=(1/n)*sum(k=0,n,(m+1)^k*C(n,k)*C(n,k-1))

a(n) = number of lattice paths from (0,0) to (n+1,n+1) that consist of steps (i,0) and (0,j) with i,j>=1 and that stay strictly below the diagonal line y=x except at the endpoints. (See Coker reference.) Equivalently, a(n) = number of marked Dyck (n+1)-paths where the vertices in the middle of each UU and each DD are available to be marked (or not): consider the original path as a Dyck path with a mark at each vertex where two horizontal (or two vertical) steps abut. If only the UU vertices are available for marking, then the counting sequence is the little Schroeder number A001003. - David Callan (callan(AT)stat.wisc.edu), Jun 07 2006

Curtis Coker, Enumerating a class of lattice paths, Disc. Math. 271 (2003) 13-28.

a(0)=1, n>0 a(n)=(1/n)*sum(k=0, n, 4^k*C(n, k)*C(n, k-1))

a(1)=1, a(n)=3*a(n-1)+sum(i=1, n-1, a(i)*a(n-i)). - Benoit Cloitre (benoit7848c(AT)orange.fr), Mar 16 2004

a(n)=Sum[1/(n+1) Binomial[n+1,k]Binomial[2n-k,n-k]3^k,{k,0,n}] - David Callan (callan(AT)stat.wisc.edu), Jun 07 2006

(PARI) a(n)=if(n<1, 1, sum(k=0, n, 4^k*binomial(n, k)*binomial(n, k-1))/n)

Cf. A006318, A047891.

Adjacent sequences: A082295 A082296 A082297 this_sequence A082299 A082300 A082301

Sequence in context: A085456 A120915 A100328 this_sequence A129378 A078944 A127088

nonn

Benoit Cloitre (benoit7848c(AT)orange.fr), May 10 2003