A110099 Number of return steps to the line y = x from the line y = x+1 (i.e., E steps from the line y = x+1 to the line y = x) in all Delannoy paths of length n.

0, 1, 8, 53, 332, 2029, 12236, 73193, 435480, 2581273, 15258256, 90005981, 530071076, 3117718213, 18318316948, 107537570513, 630844709168, 3698457841201, 21671720364056, 126932183197061, 743158103135484

Offset: 0

Author

Emeric Deutsch, Jul 11 2005

Keywords

• nonn

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).

Examples

			a(2) = 8 because in the 13 (=A001850(2)) Delannoy paths of length 2, namely DD, DN(E), DEN, N(E)D, N(E)N(E), N(E)EN, ND(E), NNE(E), END, ENN(E), ENEN, EDN and EENN, one has altogether 8 return steps to the line y = x from the line y = x+1 (shown between parentheses).
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• R. A. Sulanke, Objects counted by the central Delannoy numbers, J. Integer Seq. 6 (2003), no. 1, Article 03.1.5.

Crossrefs

Cf. A001850, A006318, A110098, A110107.

Programs

Maple

R:=(1-z-sqrt(1-6*z+z^2))/2/z: G:=z*R^3/(1-z*R^2)^2: Gser:=series(G,z=0,30): 0,seq(coeff(Gser,z^n),n=1..24);

Mathematica

CoefficientList[Series[x*((1-x-Sqrt[1-6*x+x^2])/2/x)^3/(1-x*((1-x-Sqrt[ 1-6*x+x^2])/2/x)^2)^2, {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 24 2012 *)

Formula

a(n) = Sum_{k=0..n} k*A110098(n,k).

G.f.: z*R^3/(1 - z*R^2)^2, where R = 1 + z*R + z*R^2 is the g.f. of the large Schroeder numbers (A006318).

Recurrence: n*a(n) = 3*(4*n-3)*a(n-1) - 19*(2*n-3)*a(n-2) + 3*(4*n-9) * a(n-3) - (n-3)*a(n-4). - Vaclav Kotesovec, Oct 24 2012

a(n) ~ 1/8*(2+sqrt(2))*(3+2*sqrt(2))^n. - Vaclav Kotesovec, Oct 24 2012