A109980 Number of Delannoy paths of length n with no (1,1)-steps on the line y=x.

1, 2, 8, 36, 172, 852, 4324, 22332, 116876, 618084, 3296308, 17702412, 95627580, 519170004, 2830862532, 15494401116, 85091200620, 468692890308, 2588521289812, 14330490031020, 79509491551772, 442019710668852

Offset: 0

Author

Emeric Deutsch, Jul 06 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).

Equals left border of triangle A152250 and INVERTi transform of A001850, the Delannoy numbers: (1, 3, 13, 63, 321, ...). - Gary W. Adamson, Nov 30 2008

Hankel transform is A036442. First column of Riordan array ((1-x)/(1+x), x/(1+3x+2x^2))^{-1}. - Paul Barry, Apr 27 2009

Examples

			a(2)=8 because we have NDE, EDN, NENE, NEEN, ENNE, ENEN, NNEE and EENN.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Robert A. Sulanke, Objects Counted by the Central Delannoy Numbers, Journal of Integer Sequences, Volume 6, 2003, Article 03.1.5.

Crossrefs

First column of A109979.

Cf. A152250.

Programs

Maple

g:=1/(z+sqrt(1-6*z+z^2)): gser:=series(g,z=0,28): 1,seq(coeff(gser,z^n),n=1..25);

Mathematica

CoefficientList[Series[1/(x+Sqrt[1-6*x+x^2]), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 18 2012 *)

Formula

G.f.: 1/(z + sqrt(1 - 6*z + z^2)).

Moment representation: a(n) = 0^n/3 + (1/Pi)*Integral_{x=3-2*sqrt(2)..3+2*sqrt(2)} x^n*sqrt(-x^2+6x-1)/(x*(6-x)) dx. - Paul Barry, Apr 27 2009

From Gary W. Adamson, Aug 23 2011: (Start)

a(n) is the upper left term in M^n, M = an infinite square production matrix as follows:

2, 2, 0, 0, 0, 0, ...

2, 1, 2, 0, 0, 0, ...

2, 1, 1, 2, 0, 0, ...

2, 1, 1, 1, 2, 0, ...

2, 1, 1, 1, 1, 2, ...

... (End)

D-finite with recurrence: n*a(n) = 3*(4*n-3)*a(n-1) - (37*n-57)*a(n-2) + 6*(n-3)*a(n-3). - Vaclav Kotesovec, Oct 18 2012

a(n) ~ 2^(1/4) * (1 + sqrt(2))^(2*n+3) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Oct 18 2012, simplified Dec 24 2017