0,2

Number of Delannoy paths of length n that do not start with a (1, 1) step (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)). Example: a(1) = 2 because we have NE and EN. Column 0 of A110169 (also nonzero entries in each column of A110169).

For n > 0: a(n) = A128966(2*n,n). - Reinhard Zumkeller, Jul 20 2013

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

Thomas Selig, Combinatorial aspects of sandpile models on wheel and fan graphs, arXiv:2202.06487 [math.CO], 2022.

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

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

a(n) = P_n(3) - P_{n-1}(3) (n >= 1), where P_j is j-th Legendre polynomial.

From Paul Barry, Oct 18 2009: (Start)

G.f.: (1-x)/(1-x-2x/(1-x-x/(1-x-x/(1-x-x/(1-... (continued fraction);

G.f.: 1/(1-2x/((1-x)^2-x/(1-x/((1-x)^2-x/(1-x/((1-x)^2-x/(1-... (continued fraction);

a(n) = Sum_{k = 0..n} (0^(n + k) + C(n + k - 1, 2k - 1)) * C(2k, k) = 0^n + Sum_{k = 0..n} C(n + k - 1, 2k - 1) * C(2k, k). (End)

D-finite with recurrence: n*(2*n-3)*a(n) = 2*(6*n^2-12*n+5)*a(n-1) - (n-2)*(2*n-1)*a(n-2). - Vaclav Kotesovec, Oct 18 2012

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

a(n) = A277919(2n,n). - John P. McSorley, Nov 23 2016

a(n) = 2*hypergeom([1 - n, -n], [1], 2) for n>0. - Peter Luschny, May 22 2017

D-finite with recurrence: n*a(n) +(-7*n+5)*a(n-1) +(7*n-16)*a(n-2) +(-n+3)*a(n-3)=0. - R. J. Mathar, Jan 15 2020

a(0) = 1; a(n) = (2/n) * Sum_{k=0..n-1} (n^2-k^2) * a(k). - Seiichi Manyama, Mar 28 2023

G.f.: Sum_{n >= 0} binomial(2*n, n)*x^n/(1 - x)^(2*n) = 1 + 2*x + 10*x^2 + 50*x^3 + .... - Peter Bala, Apr 17 2024

with(orthopoly): a:=proc(n) if n=0 then 1 else P(n, 3)-P(n-1, 3) fi end: seq(a(n), n=0..25);

a := n -> `if`(n=0, 1, 2*hypergeom([1 - n, -n], [1], 2)):

seq(simplify(a(n)), n=0..24); # Peter Luschny, May 22 2017

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

(PARI) x='x+O('x^66); Vec((1-x)/sqrt(1-6*x+x^2)) \\ Joerg Arndt, May 16 2013

(Haskell)

a110170 0 = 1

a110170 n = a128966 (2 * n) n -- Reinhard Zumkeller, Jul 20 2013

Cf. A001850, A110169.

Cf. A085362, A162478, A359758, A360132.

Sequence in context: A015949 A020699 A020729 * A026332 A027908 A206637

Adjacent sequences: A110167 A110168 A110169 * A110171 A110172 A110173

nonn,easy

Emeric Deutsch, Jul 14 2005

approved