A110184 Number of (1,1)-steps on the lines y=x, y=x+1 and y=x-1 in all Delannoy paths of length n.

0, 1, 8, 55, 354, 2205, 13484, 81523, 489158, 2919481, 17356752, 102884271, 608460330, 3591886293, 21172419444, 124649246955, 733107494286, 4307974826097, 25296523200920, 148448166035239, 870665283937010

Offset

0,3

Comments

A Delannoy path of length n is a path from (0,0) to (n,n), consisting of steps E=(1,0), N=(0,1) and D=(1,1).

Links

Table of n, a(n) for n=0..20.

Robert A. Sulanke, Objects Counted by the Central Delannoy Numbers, Journal of Integer Sequences, Volume 6, 2003, Article 03.1.5.

Formula

a(n) = sum(k*A110183(n,k),k=0..n).

G.f.: z(1-z-zr+z^2+z^2*r)/[(1-6z+z^2)(1-3z+z^2-zr+z^2*r)], where r=1+zr+zr^2=[1-z-sqrt(1-6z+z^2)]/(2z) is the g.f. of the large Schroeder numbers (A006318).

D-finite with recurrence n*a(n) +(-13*n+11)*a(n-1) +10*(5*n-9)*a(n-2) +10*(-5*n+16)*a(n-3) +(13*n-54)*a(n-4) +(-n+5)*a(n-5)=0. - R. J. Mathar, Jul 24 2022

Example

a(3)=55 because on the 63 (=A001850(3)) Delannoy paths of length 3 we have altogether A108666(3)=57 D-steps; however 2 of these, namely the D's in NNDEE and EEDNN, are not on the lines y=x, y=x+1, y=x-1.

Maple

r:=(1-z-sqrt(1-6*z+z^2))/2/z: G:=z*(1-z-z*r+z^2+z^2*r)/(1-6*z+z^2)/(1-3*z+z^2-z*r+z^2*r): Gser:=series(G, z=0, 27): 0, seq(coeff(Gser, z^n), n=1..24);

Crossrefs

Cf. A001850, A108666, A110183.

Sequence in context: A373116 A244501 A026994 * A013698 A154245 A143420

Adjacent sequences: A110181 A110182 A110183 * A110185 A110186 A110187

Keyword

nonn

Author

Emeric Deutsch, Jul 14 2005

Status

approved