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A109980 Number of Delannoy paths of length n with no (1,1)-steps on the line y=x (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)). 7
1, 2, 8, 36, 172, 852, 4324, 22332, 116876, 618084, 3296308, 17702412, 95627580, 519170004, 2830862532, 15494401116, 85091200620, 468692890308, 2588521289812, 14330490031020, 79509491551772, 442019710668852 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Equals left border of triangle A152250 and INVERTi transform of A001850, the Delannoy numbers: (1, 3, 13, 63, 321,...). [From 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}. [From Paul Barry, Apr 27 2009]

REFERENCES

R. A. Sulanke, Objects counted by the central Delannoy numbers, J. of Integer Sequences, 6, 2003, Article 03.1.5.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

FORMULA

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

Moment representation: a(n)=(1/pi)int(x^n*sqrt(-x^2+6x-1)/(x(6-x)),x,3-2*sqrt(2),3+2*sqrt(2))+0^n/3. [From Paul Barry, Apr 27 2009]

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,...

  ... - Gary W. Adamson, Aug 23 2011

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

EXAMPLE

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

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 *)

CROSSREFS

First column of A109979.

A152250 [From Gary W. Adamson, Nov 30 2008]

Sequence in context: A206902 A275752 A084868 * A186338 A190862 A110837

Adjacent sequences:  A109977 A109978 A109979 * A109981 A109982 A109983

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Jul 06 2005

STATUS

approved

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Last modified February 24 18:24 EST 2018. Contains 299628 sequences. (Running on oeis4.)