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A110170 First differences of the central Delannoy numbers (A001850). 5
1, 2, 10, 50, 258, 1362, 7306, 39650, 217090, 1196834, 6634890, 36949266, 206549250, 1158337650, 6513914634, 36718533570, 207412854786, 1173779487810, 6653482333450, 37770112857074, 214694383882498, 1221832400430482, 6961037946938250, 39697830840765090, 226596964146630658 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

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.

FORMULA

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)

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

MAPLE

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

MATHEMATICA

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

PROG

(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

CROSSREFS

Cf. A001850, A110169.

Sequence in context: A015949 A020699 A020729 * A026332 A027908 A206637

Adjacent sequences:  A110167 A110168 A110169 * A110171 A110172 A110173

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Jul 14 2005

STATUS

approved

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Last modified February 19 03:37 EST 2018. Contains 299330 sequences. (Running on oeis4.)