%I #24 Dec 29 2014 11:39:47

%S 0,5,44,321,2184,14325,91860,580097,3622928,22437477,138049020,

%T 844881345,5148375192,31258302933,189199514532,1142148091905,

%U 6878977097760,41347348295877,248082231062988,1486116788646977

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

%C 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).

%H Reinhard Zumkeller, <a href="/A109984/b109984.txt">Table of n, a(n) for n = 0..1000</a>

%H Robert A. Sulanke, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL6/Sulanke/delannoy.html">Objects Counted by the Central Delannoy Numbers</a>, Journal of Integer Sequences, Volume 6, 2003, Article 03.1.5.

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

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

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

%F 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

%F a(n) ~ sqrt(24 + 17*sqrt(2))*(3 + 2*sqrt(2))^n*sqrt(n)/(4*sqrt(Pi)). - _Vaclav Kotesovec_, Oct 18 2012

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

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

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

%o (Haskell)

%o a109984 = sum . zipWith (*) [0..] . a109983_row

%o -- _Reinhard Zumkeller_, Nov 18 2014

%Y Cf. A109983, A001850.

%K nonn

%O 0,2

%A _Emeric Deutsch_, Jul 07 2005