%I A109983
%S A109983 1,0,1,2,0,0,1,6,6,0,0,0,1,12,30,20,0,0,0,0,1,20,90,140,70,0,0,0,0,0,1,
%T A109983 30,210,560,630,252,0,0,0,0,0,0,1,42,420,1680,3150,2772,924,0,0,0,0,0,
               0,
%U A109983 0,1,56,756,4200,11550,16632,12012,3432,0,0,0,0,0,0,0,0,1,72,1260,9240
%N A109983 Triangle read by rows: T(n,k) (0<=k<=n) is the number of Delannoy paths 
               of length n, having k steps (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 A109983 Row n has 2n+1 terms, the first n of which are 0. Row sums are the central 
               Delannoy numbers (A001850). Column sums are the central trinomial 
               coefficients (A002426) T(n,2n)=binomial(2n,n) (A000984). T(n,k)=A104684(n,
               2n-k). sum(k*T(n,k),k=0..n)=A109984(n)
%D A109983 R. A. Sulanke, Objects counted by the central Delannoy numbers, J. of 
               Integer Sequences, 6, 2003, Article 03.1.5.
%F A109983 T(n, k)=binomial(n, 2n-k)binomial(k, n). G :=1/sqrt[(1-tz)^2-4zt^2].
%e A109983 T(2,3)=6 because we have DNE, DEN, NED, END, NDE and EDN.
%e A109983 Triangle begins
%e A109983 .1;
%e A109983 .0,1,2;
%e A109983 .0,0,1,6,6;
%e A109983 .0,0,0,1,12,30,20;
%p A109983 T:=(n,k)->binomial(n,2*n-k)*binomial(k,n):for n from 0 to 8 do seq(T(n,
               k),k=0..2*n) od; # yields sequence in triangular form
%Y A109983 Cf. A001850, A002426, A000984, A104684, A109984.
%Y A109983 Sequence in context: A138497 A113129 A127826 this_sequence A093492 A128771 
               A139380
%Y A109983 Adjacent sequences: A109980 A109981 A109982 this_sequence A109984 A109985 
               A109986
%K A109983 nonn,tabf
%O A109983 0,4
%A A109983 Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 07 2005