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

%I A109984 #24 Dec 29 2014 11:39:47
%S A109984 0,5,44,321,2184,14325,91860,580097,3622928,22437477,138049020,
%T A109984 844881345,5148375192,31258302933,189199514532,1142148091905,
%U A109984 6878977097760,41347348295877,248082231062988,1486116788646977
%N A109984 a(n) = number of steps in all Delannoy paths of length n.
%C A109984 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 A109984 Reinhard Zumkeller, <a href="/A109984/b109984.txt">Table of n, a(n) for n = 0..1000</a>
%H A109984 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 A109984 a(n) = sum_{k=0..2n} k*A109983(k).
%F A109984 a(n) = sum_{k=n..2*n} k*binomial(n, 2*n-k)*binomial(k, n).
%F A109984 G.f.: z*(5-z)/(1-6*z+z^2)^(3/2).
%F A109984 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 A109984 a(n) ~ sqrt(24 + 17*sqrt(2))*(3 + 2*sqrt(2))^n*sqrt(n)/(4*sqrt(Pi)). - _Vaclav Kotesovec_, Oct 18 2012
%e A109984 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 A109984 a:=n->add(k*binomial(n,2*n-k)*binomial(k,n),k=n..2*n): seq(a(n),n=0..23);
%t A109984 CoefficientList[Series[x*(5-x)/(1-6*x+x^2)^(3/2), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 18 2012 *)
%o A109984 (Haskell)
%o A109984 a109984 = sum . zipWith (*) [0..] . a109983_row
%o A109984 -- _Reinhard Zumkeller_, Nov 18 2014
%Y A109984 Cf. A109983, A001850.
%K A109984 nonn
%O A109984 0,2
%A A109984 _Emeric Deutsch_, Jul 07 2005