cp's OEIS Frontend

A110099 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.

%I A110099 #22 Jul 23 2017 12:18:32
%S A110099 0,1,8,53,332,2029,12236,73193,435480,2581273,15258256,90005981,
%T A110099 530071076,3117718213,18318316948,107537570513,630844709168,
%U A110099 3698457841201,21671720364056,126932183197061,743158103135484
%N A110099 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 A110099 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 A110099 Vincenzo Librandi, <a href="/A110099/b110099.txt">Table of n, a(n) for n = 0..200</a>
%H A110099 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 A110099 a(n) = Sum_{k=0..n} k*A110098(n,k).
%F A110099 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 A110099 Recurrence: n*a(n) = 3*(4*n-3)*a(n-1) - 19*(2*n-3)*a(n-2) + 3*(4*n-9) * a(n-3) - (n-3)*a(n-4). - _Vaclav Kotesovec_, Oct 24 2012
%F A110099 a(n) ~ 1/8*(2+sqrt(2))*(3+2*sqrt(2))^n. - _Vaclav Kotesovec_, Oct 24 2012
%e A110099 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 A110099 R:=(1-z-sqrt(1-6*z+z^2))/2/z: G:=z*R^3/(1-z*R^2)^2: Gser:=series(G,z=0,30): 0,seq(coeff(Gser,z^n),n=1..24);
%t A110099 CoefficientList[Series[x*((1-x-Sqrt[1-6*x+x^2])/2/x)^3/(1-x*((1-x-Sqrt[ 1-6*x+x^2])/2/x)^2)^2, {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 24 2012 *)
%Y A110099 Cf. A001850, A006318, A110098, A110107.
%K A110099 nonn
%O A110099 0,3
%A A110099 _Emeric Deutsch_, Jul 11 2005