cp's OEIS Frontend

A110123 Triangle read by rows: T(n,k) is the number of Delannoy paths of length n, having k EE's and NN's crossing the line y = x (i.e., two consecutive E steps from the line y = x+1 to the line y = x-1 or two consecutive N steps from the line y = x-1 to the line y = x+1).

%I A110123 #11 Jul 23 2017 03:03:41
%S A110123 1,3,11,2,45,16,2,197,100,22,2,903,576,174,28,2,4279,3206,1202,266,34,
%T A110123 2,20793,17568,7732,2128,376,40,2,103049,95592,47676,15452,3408,504,
%U A110123 46,2,518859,518720,286156,105528,27500,5096,650,52,2,2646723,2813514
%N A110123 Triangle read by rows: T(n,k) is the number of Delannoy paths of length n, having k EE's and NN's crossing the line y = x (i.e., two consecutive E steps from the line y = x+1 to the line y = x-1 or two consecutive N steps from the line y = x-1 to the line y = x+1).
%C A110123 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 A110123 Row 0 has one term; row n has n terms (n > 0).
%C A110123 Row sums are the central Delannoy numbers (A001850).
%C A110123 Column 0 yields the little Schroeder numbers (A001003).
%H A110123 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 A110123 Sum_{k=0..n-1} k*T(n,k) = 2*A110127(n).
%F A110123 G.f.: (1 - tzR + zR)/(1 - z - tzR + tz^2*R - zR - z^2*R), where R = 1 + zR + zR^2 = (1 - z - sqrt(1 - 6z + z^2))/(2z) is the g.f. of the large Schroeder numbers (A006318).
%e A110123 T(2,1)=2 because we have NEEN and ENNE.
%e A110123 Triangle begins:
%e A110123     1;
%e A110123     3;
%e A110123    11,   2;
%e A110123    45,  16,   2;
%e A110123   197, 100,  22,   2;
%p A110123 R:=(1-z-sqrt(1-6*z+z^2))/2/z: G:=simplify((1-z*R*t+z*R)/(1-z-z*R*t+z^2*R*t-z*R-z^2*R)): Gser:=simplify(series(G,z=0,14)): P[0]:=1: for n from 1 to 10 do P[n]:=coeff(Gser,z^n) od: 1; for n from 1 to 10 do seq(coeff(t*P[n],t^k),k=1..n) od; # yields sequence in triangular form
%Y A110123 Cf. A001003, A001850, A006318, A100127, A110121.
%K A110123 nonn,tabf
%O A110123 0,2
%A A110123 _Emeric Deutsch_, Jul 13 2005