A109984 a(n) = number of steps in all Delannoy paths of length n.

0, 5, 44, 321, 2184, 14325, 91860, 580097, 3622928, 22437477, 138049020, 844881345, 5148375192, 31258302933, 189199514532, 1142148091905, 6878977097760, 41347348295877, 248082231062988, 1486116788646977

Offset

0,2

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).

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

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

Formula

a(n) = sum_{k=0..2n} k*A109983(k).

a(n) = sum_{k=n..2*n} k*binomial(n, 2*n-k)*binomial(k, n).

G.f.: z*(5-z)/(1-6*z+z^2)^(3/2).

Recurrence: (n-1)*(2*n-9)*a(n) = 4*(3*n^2-15*n+7)*a(n-1) - (n-1)*(2*n-7)*a(n-2). - Vaclav Kotesovec, Oct 18 2012

a(n) ~ sqrt(24 + 17*sqrt(2))*(3 + 2*sqrt(2))^n*sqrt(n)/(4*sqrt(Pi)). - Vaclav Kotesovec, Oct 18 2012

Example

a(1)=5 because in the 3 (=A001850(1)) Delannoy paths of length 1, namely D, NE and EN, we have altogether five steps.

Maple

a:=n->add(k*binomial(n, 2*n-k)*binomial(k, n), k=n..2*n): seq(a(n), n=0..23);

Mathematica

CoefficientList[Series[x*(5-x)/(1-6*x+x^2)^(3/2), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 18 2012 *)

Prog

(Haskell)
a109984 = sum . zipWith (*) [0..] . a109983_row
-- Reinhard Zumkeller, Nov 18 2014

Crossrefs

Cf. A109983, A001850.

Sequence in context: A271118 A268762 A068311 * A247776 A227640 A096355

Adjacent sequences: A109981 A109982 A109983 * A109985 A109986 A109987

Keyword

nonn

Author

Emeric Deutsch, Jul 07 2005

Status

approved