cp's OEIS Frontend

A110221 Triangle read by rows: T(n,k) (0<=k<=floor(n/2)) is the number of Delannoy paths of length n, having k ED's.

%I A110221 #8 Jan 02 2015 11:11:50
%S A110221 1,3,11,2,45,18,195,120,6,873,720,90,3989,4110,870,20,18483,22806,
%T A110221 6930,420,86515,124264,49560,5320,70,408105,668520,331128,52920,1890,
%U A110221 1936881,3562830,2111760,456120,29610,252,9238023,18850590,13020480,3575880
%N A110221 Triangle read by rows: T(n,k) (0<=k<=floor(n/2)) is the number of Delannoy paths of length n, having k ED's.
%C A110221 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 A110221 Row n has 1+floor(n/2) terms. Row sums are the central Delannoy numbers (A001850). Column 0 yields A026375. Sum(k*T(n,k),k=0..floor(n/2))=2*A002695(n).
%H A110221 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 A110221 G.f.: 1/(1-z-2tz^2*R-2zR+2z^2*R), where R=[1-z-sqrt(1-6z+5z^2-4tz^2)]/[2z(1-z+tz)].
%e A110221 T(2,1)=2 because we have NED and EDN.
%e A110221 Triangle begins:
%e A110221 1;
%e A110221 3;
%e A110221 11,2;
%e A110221 45,18;
%e A110221 195,120,6;
%p A110221 R:=(1-z-sqrt(1-6*z+5*z^2-4*z^2*t))/2/z/(1-z+t*z): G:=1/(1-z-2*t*z^2*R-2*z*R+2*z^2*R): Gser:=simplify(series(G,z=0,15)): P[0]:=1: for n from 1 to 12 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 12 do seq(coeff(t*P[n],t^k),k=1..1+floor(n/2)) od; # yields sequence in triangular form
%Y A110221 Cf. A001850, A026375, A002695.
%K A110221 nonn,tabf
%O A110221 0,2
%A A110221 _Emeric Deutsch_, Jul 20 2005