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%I #18 Jul 23 2017 03:10:39
%S 0,0,1,10,75,508,3277,20566,126871,773688,4679769,28136546,168395235,
%T 1004239156,5971820709,35429993390,209800355631,1240361694064,
%U 7323260678065,43187703202234,254439363998587,1497730375793004
%N Number of EE's crossing the line y = x (i.e., two consecutive E steps from the line y = x+1 to the line y = x-1) 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).
%C {A110127}[n+2] = conv({0, {A002002})[n]. - _Tilman Neumann_, Feb 05 2009
%H Vincenzo Librandi, <a href="/A110127/b110127.txt">Table of n, a(n) for n = 0..200</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..floor(n/2)} k*A110121(n,k).
%F G.f.: z^2*R^2/(1-6z+z^2), where R = 1+zR+zR^2 = [1-z-sqrt(1-6z+z^2)]/(2z) is the g.f. of the large Schroeder numbers (A006318).
%F Recurrence: n*(2*n-5)*a(n) = 6*(4*n^2 - 13*n + 8)*a(n-1) - 4*(19*n^2 - 76*n + 75)*a(n-2) + 6*(4*n^2 - 19*n + 20)*a(n-3) - (n-4)*(2*n-3)*a(n-4). - _Vaclav Kotesovec_, Oct 24 2012
%F a(n) ~ 1/8*sqrt(2)*(3+2*sqrt(2))^n. - _Vaclav Kotesovec_, Oct 24 2012
%e a(2) = 1 because, among the 13 (=A001850(2)) Delannoy paths of length 2, only NEEN has an EE crossing the line y = x.
%p R:=(1-z-sqrt(1-6*z+z^2))/2/z: G:=z^2*R^2/(1-6*z+z^2): Gser:=series(G,z=0,27): 0,seq(coeff(Gser,z^n),n=1..24);
%t CoefficientList[Series[x^2*((1-x-Sqrt[1-6*x+x^2])/2/x)^2/(1-6*x+x^2), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 24 2012 *)
%Y Cf. A001850, A110121.
%K nonn
%O 0,4
%A _Emeric Deutsch_, Jul 13 2005
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