%I
%S 0,1,8,53,332,2029,12236,73193,435480,2581273,15258256,90005981,
%T 530071076,3117718213,18318316948,107537570513,630844709168,
%U 3698457841201,21671720364056,126932183197061,743158103135484
%N Number of return steps to the line y = x from the line y = x+1 (i.e., E steps from the line y = x+1 to the line y = x) 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 Vincenzo Librandi, <a href="/A110099/b110099.txt">Table of n, a(n) for n = 0..200</a>
%H R. A. Sulanke, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL6/Sulanke/delannoy.html">Objects counted by the central Delannoy numbers</a>, J. Integer Seq. 6 (2003), no. 1, Article 03.1.5.
%F a(n) = Sum_{k=0..n} k*A110098(n,k).
%F G.f.: z*R^3/(1  z*R^2)^2, where R = 1 + z*R + z*R^2 is the g.f. of the large Schroeder numbers (A006318).
%F Recurrence: n*a(n) = 3*(4*n3)*a(n1)  19*(2*n3)*a(n2) + 3*(4*n9) * a(n3)  (n3)*a(n4).  _Vaclav Kotesovec_, Oct 24 2012
%F a(n) ~ 1/8*(2+sqrt(2))*(3+2*sqrt(2))^n.  _Vaclav Kotesovec_, Oct 24 2012
%e a(2) = 8 because in the 13 (=A001850(2)) Delannoy paths of length 2, namely DD, DN(E), DEN, N(E)D, N(E)N(E), N(E)EN, ND(E), NNE(E), END, ENN(E), ENEN, EDN and EENN, one has altogether 8 return steps to the line y = x from the line y = x+1 (shown between parentheses).
%p R:=(1zsqrt(16*z+z^2))/2/z: G:=z*R^3/(1z*R^2)^2: Gser:=series(G,z=0,30): 0,seq(coeff(Gser,z^n),n=1..24);
%t CoefficientList[Series[x*((1xSqrt[16*x+x^2])/2/x)^3/(1x*((1xSqrt[ 16*x+x^2])/2/x)^2)^2, {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 24 2012 *)
%Y Cf. A001850, A006318, A110098, A110107.
%K nonn
%O 0,3
%A _Emeric Deutsch_, Jul 11 2005
